AB - cheetah.fi · AB HELSINKI UNIVERSITY OF TECHNOLOGY Department of Electrical and Communications...

AB HELSINKI UNIVERSITY OF TECHNOLOGYDepartment of Electrical and Communications EngineeringLaboratory of Computational Engineering

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5SnAsin(ωt)

x>4|DTI1JK9<=?e 7:@wh:7A@?DTJ[Ve[M\DT@HX 5gJ BEMA9J @F7AI6F@?Jh = 2mm

A = 50µm

7A4FXω = 10rad/s

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hi=?=

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fx10 l@HMA;>V]MABmy R x = Fx10; ?wBEMA9Q;UPTJDTV<JKX V<J[;>VoMABW687A9U7AIJu;>J[9UVKx ?_=HZ\DT9UJ[V]MA4 ;>PTJ@?J[B ;oeKM\@HDTI14 = 7_ep7A4FX J6? eKM\9<9UJKV<6CMA4TX ;>M";UPTJJ[RSD87:;>=HM\4TVsM:B

u7A4TX 5_X

7A4TX BN;UM;UPTJJ[RiDT7:;>=HM\4TVQMABvx 2mPTJ687:9>7AI1J[;UJK9UV]MAB ;UPTJV<niV<;UJKI JK9<J=H4

7 7A4TX+5@?Du = 1.032

Dv = 0.516

α = 0.899

7:4TXβ = −0.91

me7A4TX X2?

Du = 0.244Dv = 0.122

α = 0.398

7A4FXβ = −0.4

J17A4TX B"?Du = 0.129

Dv = 0.516

α = 0.89

7A4TXβ = −0.99

xxYxQx]xQxQx]xQx]xQxQx zK;fx z l@HMA;>VMABYy R x = fx10T ?sBEM\91;>PTJDTVUJ[XV<J[;pM:BY687A9>7:IJu;>JK9<VKx ? =?Z\DT9<JMA42;>PTJ

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;>PTJ1JKRSD87:;U=?M\4 W= ;>Pvx 2mPTJ1687A9U7AI1J[;>J[9UV JK9UJ

Du = 0.125Dv = 0.05

F = 0.045

7A4TXK = 0.065

x xQx]xQxYx]xQx]xQxQx]xQx]xQxYx]xQxQx]xQx]xQxQx]x z)S

3Fx10 2mPTJWXT=?V<6CJ[9UVU=HM\4 9UJ[@ 7:;U=?M\4sM:B y R x = 3Fx z ? BEMA9 VUJ[@?JKeu;>J[XI1MfXFJKVKx\2mPTJW687A9>7:IJu; BJK9UV JK9UJ

D = 0.516α = 0.899

β = −0.91

7A4TXδ = 2

BEM\9k = 0.45= X87AV<P.B5XFMA;p@?=H4TJ6?

D = 0.122α = 0.398

β = −0.4

7A4TXδ = 2

BEM\9k = 0.84

= XTMA;U;UJKX@?=?4FJ6?/D = 0.516

α = 0.89

β = −0.99

7A4TXδ = .25BEM\9

k = 0.96= V<M\@?=HX@H=?4TJ ? x2mPTJ 9UJ[Z\=?MA475gM h\J

λ(k) = 05gM\DT4TXFV ;>PFJ N7 hAJ

4iDFI 5CJ[9mhA7:@?DTJ[VWBEMA9mDT4TV<;>75T@HJrIMiXTJKV[x xQx]xQxQx]xQx]xQxYx]xQxQx]xQx]xQxQx]xK;

hf=H=?=

3Fx z 7:;<;>JK9<4TV MK5F;>7A=?4TJ[X BE9UM\I VU=HIsDT@ 7 ;>=?MA4TV]M:Bb;UPTJpZ\J[4TJK9U7A@O2ODT9U=H4TZ IMiXTJ[@ =?4; MXT=?I1JK4TV<=?M\4FVrDTVU=H4TZ9U7A4TXTMAI-=?4T= ;>=?7A@ e[M\4TXT= ;>=HM\4TVKx 2OM\6 9<MYj

r1 = 3.57A4TXr2 = 0 5CM:;U;>MAI9UMYj r1 = 0.02

7A4TXr2 = 0.2

xN354;UPTJQ@?J[B ;be[M\@?DTI14;>PTJr687:9>7AI1J[;UJK9UV JK9<Jre PTM\V<JK4 ;>MsB 7 h\M\9

k = 0.45 =H4;>PFJr9U=?ZAPS;NeKM\@HDTI4;>MB 7 hAM\9WIMiXTJk = 0.84

x AWJK6T9<MfXTDFeKJKXBE9UM\I =N7A9<9U=HM1Ju;b7A@ xE`0TTKS! x xQx]xQxKT3Fx 7:;<;>JK9<4TV M5F; 7A=H4TJKX BE9UMAIVU=?IoDT@ 7:;U=?M\4FV MAB ;UPTJbZ\JK4TJ[9>7A@2ODT9U=H4TZ]I1MiXTJK@ 7:B ;UJK9

500 000=H;UJK9U7:;>=HM\4TV 7A4TX DTVU=H4TZr9>7:4TXTM\I =?4T= ;>=?7A@LeKM\4FXT=H;U=?M\4TV_M\4 J[hAJK9<nsVU=H;UJNMAB

;>PTJm@?7:;U;U=?eKJ:ju0, v0 ∈ (0, 1)

x2 M\6 9<MYjr1 = 3.5

7A4FXr2 = 0 5CMA;<;>M\I 9<M Yj

r1 = 0.027A4TX

r2 = 0.2x JuB ; eKMA@?DTI148;>PTJ]687A9U7AIJu;>J[9UV J[9UJse PTMAVUJK4`;UM

B 7 hAM\9k = 0.45 =?419U=HZ\PL; eKM\@HDTI14 ;UM]B 7 h\MA9lI1MiXTJ k = 0.84

x a MAI687:9UJm;>PTJ9UJKV<DT@H;UV W=H;>P ;>PFJ ; M)B XT=?I1JK4FVU=?MA487A@)687 ;U;>J[9U4TVW=H4 ? =?Z\DT9<J%3Tx zfx x]xQx]xQxQx]x 3\

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k = 0.967A4TX 4FM\4.B5@H=?4TJK7A9O687A9>7:IJu;>JK9<V

r1 = 0.027A4TX

r2 = 0.2 7:4TX 52?m7:B ;>J[9

2 000 000;U=?I1J]Va;>J[6TV

k = 0.45

r1 = 3.57A4TX

r2 = 0.2x xQx]xQxQx]xYxQx]xQx]xQxQx]xQx]xYxQx]xQxQx]xQx]xQxQx 3K3

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k = 0.45= ?_=?ZFx3Tx9852?>x\2mPTJb6T@?M:; eKM\I 5F=?4TJ[VdRiDT7A4L;>=H;>7:;>= h\J[@Hno;UPTJ

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= @

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Dv = 0.05

F = 0.045

K = 0.065

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= VUM\@H=?X @H=?4TJ ?/S7A4TXF = 0.065

K = 0.0625

= XT7AVUPTJ[X@H=?4TJ ?7A4TX

F = 0.045K = 0.065

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F = 0.065

K = .0625Du = 0.125

Dv = 0.05

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F = 0.045K = 0.065

Du = 0.125

Dv = 0.05

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=?VQV<; 7)5T@?J=?4 ;UPTJVUJ[4TVUJpM:B n\7A6TDT4FM h=HBr7 6CJ[9<;>DF9 5gJKX VanfVa;>J[I 4TJuh\JK9XTJ[hi= 7 ;>JKV5hAJK9<n B 7A9 BE9UM\I ~Xs

x 4 ;UPTJ`MA;UPTJK91P87A4FXW7V<niV<;UJKI =?V V>7A=HX;>M 5CJY7:V<niI6f;>MA;U=?e 7:@?@Hn Va; 75F@?Jr=HBO= ;NV>7 ;>=?VaG8JKVm;UPTJr6T9UJuhi=?M\DTVWVa; 7:;UJKI1JK4L; 87:4TX=?47AXTXF=H;>=HM\4 ;>PTJ 6gJK9a;>DT9 5CJ[XV<niV<;UJKI ;UJK4TXTVr;UMp9UJ[;UDT9U4 ;UM;>PTJ M\9<=?Z\=H487A@ V<;>7:;>J ~Xs

7AV ;>=HIJZ\MfJ[VrM\4)xsk|4 7:V<niI6f;>MA;U=?e 7:@?@Hn Va; 75F@?J V<niVa;>JKI =HVY7A@?V<MpV<;>75T@?J =H4`;>PTJ V<JK4TV<JMAB )nL7A6FDT4TM h@PTM Juh\JK9;>PTJsMA6T6CMAVU=H;UJo=HVw4TMA; 7A@ 7 niVw;>9<DTJAxY3 Bl7VanfVa;>J[I-=?V|4TMA;rVa; 75T@HJ<C=H;|=?V VU7A=?X";UMe5gJDT4TV<;>75T@HJAx

ci; 75F=?@?= ;5n e 7:4 5CJXFJ[G84TJ[X I7:;UPTJKI7:;U=?e 7:@?@Hn"7AVoBEMA@?@?MWV[j =HBmM\4FJXTJ[GT4TJKV 7"4FM\9UI ‖ · ‖ =H4;>PTJ|6TP87AV<J VU687:eKJr7A4FXXTJK4FMA;>J[VN;>PTJ|6CM\=H4L; WPTJ[9UJ ;UPTJwVanfVa;>J[I =?V 6CJ[9<;UDT9 5gJKX;UM ~X0

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∀ ε > 0 ∃ δ(ε) > 0 s.t. ∀ ~X0, ‖ ~X0 − ~Xs‖ < δ ⇒ ‖ ~X(t) − ~Xs‖ < ε ∀ t ≥ 0.

2mPTJYV<niV<;UJKI =?VW7AVaniI6F;UMA;>=He 7A@H@Hn1V<;>75T@HJr=HB T=H47:XTXT=H;U=?M\4; ‖ ~X(t) − ~Xs‖ → 07AV

t → ∞ x

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8

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φjE06?d3 ; 9UJKI7A=H4TV DF4Te P87A4TZAJKX

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3541;UPTJ ?g7:9>7AX87 noJ@f6CJ[9U=HIJ[4S;d;UPTJWV<niI1IJu;>9<n 5T9UJK7AtS=?4TZYe 7:4 5gJWDT4TXTJ[9UV<;UMfMiX17AVdBEM\@?@HMWVKjOkN;G89UVa; S;>PTJ[9UJw=HV 7QDT4T= BEM\9UI@?7 n\JK9lMAB);UPTJw@H=?RSDT=HXL= x JAxEi7]6gJK9<BEJ[e[;U@Hn V<niI1IJu;>9<=?ebVU=H;UD87:;U=?M\4)xOkw6VB6T@Hni=?4FZ J@i;>J[9U487A@)BEMA9UeKJ[V9= hi=65T9U7:;>=HM\4TV"? ;>M ;UPTJYV<niV<;UJKI 9UJ[VUDT@ ;>Vm=H47:47:V<niII1J[;U9U=He VU= ;>D87:;U=?M\4=?4;UPTJwV<JK4TV<J ;>P87 ; ;UPTJw@?7 nAJK9 =?V 4FMA; DF4T=HBEM\9M17A47AVanfI1I1J[;>9<=?erV<=H;>DT7:;>=HM\4V<;>7A9<;U=?4TZ BE9<M\I7 V<niI1IJu;>9U=He M\4TJ:x

0 z

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0

=9> ="> = 9M@GIK IF P D FS G KBD F

2mPTJ A|7 ni@?J[=?Z\P.B =EDK4T7A9UX`J @f6CJ[9U=?I1JK4L;Ye[M\4TeKJ[9U4TVY;>PFJ 6T9UMK5F@?JKI%MABl;>PTJ[9UI7A@_e[M\4Lh\JKeu;>=HM\4 7:4TXN7AV G89UVa;d6CJ[9<BEM\9<IJ[X 5Ln =ED[487A9<X Y0TA\! x 3 ; 7AV 4FMA; DT4L;U=?@FV<=Y@i;>J[JK4snAJ 7A9<Vd@ 7:;UJK9 WPTJK4 M\9<XA|7 ni@?J[=?Z\P ;>9<=?JKX1;>MsJ @f6T@ 7A=H41;UPTJw6FPTJK4TMAIJ[4TM\4;UPTJKM\9<J[;U=?e 7:@?@Hn 9Ab7 nf@HJK=HZ\P0T&08! x_kw@H;UPTM\DTZAP;>PTJ J @F6gJK9<=?I1JK4L; 7:@ VUJu;>DT6=H;UVUJ[@HBl=?V V<=?I16T@?J! = ;rXT=?V<6T@ 7 niVwhAJK9<n9<=?e P 5gJKP87 hi=?MA9Kxo354Juh\JK9aniX87 n@?=HBEJ!J@F7AI16T@?J[VrMAB ;UPTJK9<I7:@ eKM\4LhAJKe[;U=?M\4 9UJ[@ 7:;UJKX;>Mp;>PT=HVYJ@f6CJ[9U=HIJ[4S;]7A9<J ;>PTJ eK=H9Ue[DT@ 7:;U=?M\4MAB 7:;>J[9 =?4"7:;>I1M\VU6FPTJK9<Js7A4TX"=?4;UPTJoMieKJK7A4TV XFDTJ];UM JK7:;>PFJK9 e P87:4TZ\JKV7A4TX;>PTJ]IMA;U=?M\4MAB_e[M\4L;>=H4TJK4L; 7:@6T@ 7:;UJKVmeK7ADTVUJ[X^5LnpI1M h\JKI1JK4L;UV|MAB I7AZ\I7o=?4p;>PFJQI7A4L;>@HJ MABO;>PTJYJK7A9<;UP)x

2mPTJ|=HXTJ 7rMABAb7 ni@HJK=?ZAP.BC=ED[487A9<X eKM\4Lh\J[e[;U=?M\4=?Vd=?@H@?DTVa;>9U7:;>J[X =?4 ?_=?ZADT9UJbzfx98Fx[a M\4TVU=HXTJK9l7 78DT=HX6T@ 7Ae[JKX 5gJ[; JKJK4 ; M =?4FGT4T=H;UJ687A9U7A@?@HJK@ PFM\9U=1UKM\4L; 7:@N6T@ 7 ;>JKV 7:VUVUDFIJ[X ;UM 5gJ 6CJ[9<BEJ[e[;1PTJ 7 ;eKM\4TXFDTe[;UM\9UV[xi] JXTJ[4TMA;UJ;UPTJ;UJKI16CJ[9>7:;UDT9UJ MABb;>PFJ@HM JK9 6T@?7:;>J 5Ln

T1

7A4TX ;>PFJDT6F6CJ[96T@ 7:;UJ 5Ln

T2

7A4FX;UPTJr;>JKI16gJK9>7 ;>DT9<JQXF=1XCJK9UJ[4TeKJ5gJ[; JKJK4;>PTJY@?M JK9b7A4TXDT6T6gJK9b6T@?7:;>J[Vf5Sn∆T = T1 − T2

xm3 B∆T < 0

;>PTJ[9UJ W=?@?@5CJ]4TJ[;|IM:;>=?MA4=H4;UPTJ 78DF=?X)xm3 B∆T=HV|6gM\VU= ;>= h\J<

5TDF;1VUI7A@?@l;>PTJp;UPTJK9<I7A@ eKM\4FXTDTe[;U=?M\4 BE9UMAI @?M JK9 ;>MDT6F6CJ[916F@ 7:;UJ; 7At:JKV16T@ 7:eKJ 7A4TX ;>PFJ;>JKI16gJK9>7 ;>DT9<J|6T9<MAG8@HJ 5gJ[; JKJ[4

T17:4TX

T25gJKeKMAIJ[V @?=H4TJ 7:9Kx\2mPFJ 78DT=HX W=H@?@89UJ[I7:=?47:;l9UJKVa;

VU=?4FeKJ];UPTJohi=?V<eKM\V<=H;5n7A4TX;>PTJ[9UI7A@ eKM\4TXFDTe[;U=?M\4 7:9UJs7)5T@?JY;>MVa; 75F=?@?=1UKJr;>PFJoV<niVa;>JKI 7AZL7A=H4TV<;VUI7A@?@ 6CJ[9<;>DF9 587:;U=?M\4FVKjhi=?V<eKM\V<=H;5n Z\JK4TJ[9>7:;UJKV =?4L;UJK9U4T7A@ BE9U=He[;U=?M\4 MA6T6CMAVU=?4FZ I1M h\J[IJ[4S;17:4TXXT=?V<VU=?6T7:;>= h\JrJlXCJKeu;>VbMABO;>PTJ[9UI7A@)eKMA4TXTDTeu;>=?MA4 ;UJK4TX ;>M 9<JKV<;UM\9UJY;>PFJQXF=?VU6F@ 7Ae[JKI1JK4L;>V[x

kwVd;>PTJ∆TJ @Fe[JKJ[XTVm7reKJ[9<; 7:=?4 ;>PT9<JKVUPFM\@?X

∆Tc

\;UPTJK9<I7A@8J @f687A4TV<=?M\41e 7ADFVUJKV ;>PFJ 78DF=?X 4TJ 7:9;>PTJs@HM J[9r6T@ 7:;UJ];UM 5CJsV<DM5TVa; 7A4L;>=?7A@?@ n@?JKV<V XTJ[4TVUJ ;>PT7A4`;>PTJ e[M\@?XFJK9 7TDT=?X75gM h\J! WPT=?e P=?V7A4 DT4TVa; 75T@HJsVU= ;>D87 ;>=?MA4`=?4`ZA9>7 hi=H; 7 ;>=?MA487A@G8JK@HX)x 2mPTJ 7A9UI1JK9)7TDT=?X 4TJK7A9w;UPTJ @?M JK9 6T@?7:;>JN7A4L;UVW;UM 9U=?V<J<M5FDF;m;>PTJ[9UJY=?Vm4FM1V<687Ae[JYBEMA9m=H;W75gM h\J:x 2mPSDTVWV<M\IJrM:B ;UPTJYeKM\@HXTJK9NI1M\@HJKeKDF@?JKVIsDTV<;lI1M h\JWXTMW4XTDFJW;UM]e[M\4TV<JK9<h:7:;U=?M\41MABCI7:VUVl7A4TX1I7:tAJWVU687:eKJWBEM\9 N7:9UI1JK9 IMA@?JKe[DT@?J[VKx2mPT=?VdG&@fJKVl7rG84T= ;>J 7 hAJ|@HJK4TZ:;>P BEM\9 ;>PFJb=H4TV<;>75T=H@?=H;5n<:;UPTJ N7 h\Jb@?J[4TZA;UP1I1J 7:4T=?4TZ ;UPTJ W=?XF;UPMAB ;UPTJrBEM\9UI1JKX e[M\4Lh\JKeu;>=HM\4e[JK@H@?VKx 2mPF=?VW=?Vme[@?J 7:9U@Hn1hi=?V<=65T@HJr=?4 ?_=HZ\DT9UJ[Vbzfx9817:4TX zfxb;fx

kwV|71I1M\@?J[eKDT@HJrIM hAJKVwDF6 7A9<XTVC=H;W=?VWe[MfM\@HJKXXTMW4 7A4TX7AV|79<JKVUDF@H;b=H;WB 7A@H@?Vf587Ae tpXFM W4 xa M\4Lh\JKeu;>=HM\4e[JK@?@HVl7A9UJWe 7:@?@?J[X =EDK487:9UXe[JK@H@?V 7A4TX1;>PTJ 78DF=?X W=?@?@FV<; 7:9<;lIM hi=?4FZQDF67A4TX1XTMW4=?4]7|eK=H9Ue[DT@ 7A9B 7AV<PT=?MA4)x\? =?Z\DT9<J zfxb8WVUPFM WV 7|RSD87A@H=H; 7 ;>=HhAJ hi=?J M:B=ED[487A9UXoeKJK@H@?V[x[a @HJ 7A9<@Hn ;>PFJ9UJKV<DT@H;U=?4TZVU= ;>D87:;U=?M\4=?V 4FMA; VanfI1I1J[;>9<=?e:j 5gMA;UP";>9U7A4TVU@?7:;>=HM\487A@O7A4FX 9UMA;>7:;>=HM\487A@ V<niI1IJu;>9<=?JKV7A9UJ 5T9<M\tAJ[4 XFDTJ`;>M ;>PFJ IM h\J[IJ[4L;M:BQ;UPTJ I1M\@?J[eKDT@HJKVKx |M J[h\J[9rM\4FJ VUPTM\DF@?X 4TMA;U=?eKJ;>P87:;wBE9UM\I ;>PTJ I7Ae[9UM\V<eKM\6T=Hes6gM\=?4L;rMAB hi=?J ;UPT=?Vw=?V 4TM:; ;>PTJ eK7AVUJ!5CJ[e 7ADTV<Js;>PTJshAJK@HMfe[=H;5n;>JKI16gJK9>7 ;>DT9<Jr7A4TXpXTJ[4TVU= ;5np7:;m7oZ\=HhAJK4 6gM\=?4L;m7:9UJw;U=?I1JB5=H4TXTJ[6CJ[4TXTJK4L;Kx ?_=?ZADT9UJ zfx<;oV<PTMWVW7VU=?IoDT@ 7:;U=?M\49UJ[VUDT@ ;|M:B 7 I1MfXTJ[@)=?4L;U9UMiXTDTeKJ[X 5Lnc W=HB ;b7A4TX |MAPTJK45gJK9UZ 0T(;K;6 x

03

T1 T2

T1

?_=?Z\DF9UJozfxb8fjO2mPTJoG8ZADT9UJ]VUPTMWV RSD87A@H=H;>7:;>= h\JK@ n;>PFJo=?XFJ 7M:B A|7 ni@?J[=?Z\P.B =EDK4T7A9UXeKM\4LhAJKe[;U=?M\4 x=EDK487:9UX eKJ[@?@?V 7:9UJ`BEM\9<IJ[X 7A4TX ;>PTJ 7TDT=?X V<;>7A9<;UV;>M I1M hAJ =?4 eK=H9Ue[DT@ 7A91B 7AVUPF=?M\4 7A4FX ;UMM\6T6gM\VU= ;>J XT=?9<JKeu;>=?MA4TVY=?4 4FJK=?ZAPV5gM\9<PTMfMiXT=H4TZpeKJ[@?@?V[xe2mPTJVU=H;UD87:;U=?M\4VUPFM W4 =H4 ;UPTJ G8Z\DT9<J=?V=?XTJK7A@7A4TX =H49<J 7A@H=H;5n;>PTJYeKJ[@?@HVm7A9UJY4TM:;b7A@ 7 nfVW=H46T7A9>7A@H@?J[@)7A4TXXTJuBEJKe[;U@?J[VUVKx

?_=?Z\DF9UJ zfxb;fjHa M\I16TDF;UJK9 V<=?IsDT@?7:;>=HM\4 M:B%Ab7 nf@HJK=HZ\P.BC= DK487A9<X eKM\4LhAJKe[;U=?M\4 DTVU=H4TZ`;>PFJ"c W=HB ; B|M\PTJ[45CJ[9UZ IMiXTJ[@ x[2mPTJ hi=HJ =?V BE9<M\I 7)5CM h\J:x 4|MA;U=?eKJ|;>P87:; ;>PTJre[M\4Lh\JKeu;>= h\JQeKJK@H@?Vm7:9UJ 4TM:;M\9UXTJ[9UJ[X 7AVN=?41;>PTJw=?XTJK7A@CV<=H;UD87:;>=HM\4M:BQ?_=?Z\DF9UJwzfxb8fx[2mPTJrX87A9<t7:4TX@?=HZ\PL; e[M\@?MA9UV XFJK4TMA;UJw;>PFJXTM\I7A=?4TV W=H;>PDT6 7A9UX IM hAJKI1JK4L; = 7A9UI1JK9rV<687A9UV<J@H=?RSDT=HX2?W7A4FX XTMW4 7A9UX IM h\J[IJ[4L;= eKMiM\@HJK9 XTJK4FVUJw@H=?RSDT=HX2?/L9<JKVU6gJKeu;>= h\JK@ n\x 2mPTJbG8Z\DT9<Jw=HVl9<JK6T9<MfXFDTeKJ[XBE9UM\I qY7A9<;<;>DT4FJK4FzA\&0 x

07

Ju;oDFVo4TM/7A6T6F9UML7Ae P ;>PTJJ@f6gJK9U=HIJ[4L;]BE9<M\I7I1M\9<J1;UPTJKM\9<J[;U=?e 7:@l6gJK9<VU6gJKeu;>=HhAJ7A4FX XTJ B;>JK9<I=H4TJp7è[M\4L;>9UMA@N687A9U7AIJu;>J[9sXTJKV<eK9<=65T=H4TZ`;>PTJp=H4TV<;>75T=H@?=H;5nAx 2mPT=?V e[M\4L;>9<M\@N687A9U7AI1J[;>J[9s=?V7"XT=?I1JK4FVU=?MA4T@?J[VUVQ4iDFI 5CJ[9 WPT=He P e 7A6f;>DT9<JKVo;UPTJJKV<VUJ[4S;U= 7A@lBEJ 7:;UDT9UJ[VsMABm;UPTJV<niVa;>JKIx ] JW=?@?@ 4TM:;N9U=HZ\M\9<M\DTVU@ n XTJK9<=HhAJ|;>PFJ A|7 ni@?J[=?Z\P4SDTI 5gJK9

Ri= x JAxEf;UPTJ e[M\4L;>9UMA@C687:9>7AI1J[;UJK9 M:B);>PT=HV

687A9<;U=?e[DT@ 7A9QV<niVa;>JKI 5TDF; J W=?@?@ ; 7:tAJ 7`@?MiM\t 7:;s;UPTJpI1M\V<; =HI6gM\9a; 7A4L;]RiDT7A4L;>=H;U=?J[Vo;>P87 ;7)X J[e[;W;>PFJZAb7 ni@?JK=HZ\P.B =EDK487:9UXp=?4FV<; 7)5T=?@H=H;5n7A4TX ;>PTJAb7 nf@HJK=HZ\Pp4SDTI 5gJK9Kx

2mPTJ";>PFJK9UI7A@WJ@f687:4TVU=HM\4 XT=HVUeKDFVUVUJ[X 75gM hAJ`=?V1MABre[M\DT9<VUJ`=HI6gM\9<;>7A4L; x k|V∆T

=?V1VUI7A@?@ JeK7A4 7A6F6T9UM@f=?I7:;>Je P87A4TZAJ=?4 ;>PTJXTJK4TV<=H;5n

ρ= M\9]I7:VUV3? DTVU=H4TZ"@H=?4TJK7A9Q9UJKV<6CMA4TVUJ!d= x JAxE

ρ = ρ0 + ρα(T2 − T1) WPTJ[9UJ

α=HVN;>PTJ ;UPTJK9<I7A@CJ@f687A4FVU=?MA4e[MfJ1eK=?J[4L; x 354p;>PTJY7A4T7A@HniVU=HV

;>PTJrXT=HV<;>7A4TeKJrM:BO;UPTJr6T@?7:;>J[VhTZ\9>7 hi=H;>7:;>=HM\487A@g7Ae[eKJ[@?JK9U7:;>=HM\4

g7A4TXphi=?V<eKM\V<=H;5nMAB ;>PTJ 78DF=?X

νXTJ[G84FJr7A@?V<Moe P87:9>7Aeu;>JK9<=?Va;>=?e[V MAB ;>PTJ|V<niV<;UJKIxl3547AXTXT= ;>=?MA4(5LnDTV<=?4TZ];>PTJ[9UI7A@ eKM\4TXFDTe[;U=Hhi=H;5nk7A4TXVU6gJKeK= G8ePTJ 7 ;

Cp

MABQ;>PTJ*78DT=HX7:; eKM\4FV<; 7:4S;6T9<JKV<VUDT9<J< JeK7A4 XFJ[G84TJ;>PFJK9UI7A@XT=1XCDTVU= hi=H;5n

κ = k/ρCp

x ] = ;>P"PFJK@?6"MAB ;>PFJKVUJoe P87A9>7:e[;>J[9U=HV<;>=HeKVb;>PTJ A|7 ni@?J[=?Z\P4iDFI 5CJ[9weK7A45CJYXTJ[9U=HhAJKXH5LnpXT=?I1JK4FVU=?MA4 7A487A@ nfV<=?V WPT=?e P ni=?J[@?XTV

R =αg∆Th3

κν

= zix ?

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ut = Du∇2u + f(u, v)

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XTM\I7A=?4 = x J:xBlPTMI7A4Ln Z\9<=?X VU= ;>J[V7A9<JpDTVUJ[X2=H4 JK7Ae P2XF=?9UJ[e[;U=?M\4)x ?8M\9 V<=?I16T@?=HeK= ;5n ;>PFJJKRSD87:;U=?M\4TVs7A9UJDTVUD87:@?@Hn V<M\@HhAJKX =?4 7"9UJ[e[; 7:4TZ\DT@?7A9oI1JKV<P)x 2mPTJ M\9UXTV C/ 3 " 7:4TX.!+3 7:9UJDTV<JKX BEM\9Q;>PTJXF=?VUe[9UJu;>=6U[JKX XFM\I7:=?4 7:4TX ;>PTJ M\9<X " IJK7A4TV]M\4TJe[JK@?@ MABm;>PFJIJ[VUP)x

0T

3 ;m=?Vm7oBEDF4TX87AI1JK4L; 7:@CB 7Aeu;N;>P87 ;N;>PTJr6F9UMK5T@HJKI/MABOy Rgx= FxY0!?d=HVN4TMA; J[@?@YB 6CMAVUJKX W= ;>PTMADF;N;>PFJti4FM W@HJKXTZAJ MAB;>PFJ =?4T= ;>=?7A@ 7A4TX 5gM\DT4TXT7A9<neKM\4FXT=H;U=?M\4TV[x 354p687:;<;>J[9U4BEM\9<I7:;U=?M\41VU=HIsDT@?7:;>=HM\4TV=?4T= ;>= 7:@Oe[M\4TXT= ;>=HM\4TV 7:9UJ DTV<D87A@?@ n"9U7A4TXTMAI%6gJK9a;>DT9 587:;>=HM\4TVr7:9UM\DT4FX;UPTJ V<;UJ 7AXfnB V<;>7:;>J MAB ;>PFJV<niV<;UJKI (5CJ[e 7ADTV<JN=HB XF=1XCJK9UJ[4L; =?4F=H;>=?7A@LeKMA4TXT=H;U=?M\4FV 9<JKVUDF@H;d=H4s687:;<;>JK9<4TV_;UP87:;d7:9UJ ;UM\6gM\@?M\ZA=?e 7:@?@Hn;>PTJ`VU7AI1J<m=H;6T9<M hAJKV;UPTJ`9UM5TDTV<;U4TJKV<VpMABr;>PFJ`V<niV<;UJKIx 2mPTJ 5gM\DT4TXT7A9<n eKM\4FXT=H;U=?M\4TV7A9<JMAB ;>J[4"e PTM\V<JK4`;UM:5gJo6gJK9U=HMfXF=?eY=?4"7A@?@);UPTJoe[MfM\9<XT=?4T7:;>JYXT=H9UJKeu;>=HM\4TVbMAB ;>PFJoV<niVa;>JKIx _JK9<=?MiXT=?e5CM\DF4TX87A9anèKMA4TXT=H;U=?M\4FVrIJK7A4`;>P87 ;r;>PTJ M\6T6gM\V<=H;>J 5CM\DF4TX87A9<=?JKVrM:B ;UPTJ VU=?IoDT@ 7:;U=?M\4XTM\I7A=?47A9UJ]7Ae[;UD87A@H@Hnp4TJ @i;w;>MJ 7Ae P"MA;>PTJ[9 BE9UM\I WPT=?e P`= ;WBEM\@?@HM WVm;UP87:;b;>PTJ]V<niV<;UJKI%7Ae[;UVw7AV|=HB_=H;>VXT=?I1JK4TV<=?M\4FV JK9<J =?4FG84F=H;>J]6T9UM hi=?XFJKX`;>PT7:;w;UPTJsV<niVa;>JKI =?Vw4FMA;w;UMfMV<I7:@?@ x#] JsI7 n ;>PF=?4TtMAB;>PTJrV<=H;UD87:;>=HM\4p7AV ;>PTJ[9UJ M\DT@?X 5gJr7A4pDF4T@?=HI= ;>JKX7AI1M\DT4L;mMABOV<=?IsDT@?7:;>=HM\41XTM\I7A=?4TV W=H;UP;>PTJ[=?9 V<=?XTJ[VQe[M\=?4FeK=?XF=?4TZTxQ354;>PTJ eK7AVUJ MAB 7pVUI7A@?@OV<niV<;UJKI 6gJK9<=?MiXT=?e 5gM\DT4FX87A9<n"eKM\4FXT=H;U=?M\4TVI7 nXF=?V<;UM\9<;m;UPTJrG8JK@HXTVW=?4TV<=?XTJ ;UPTJYV<niV<;UJKIx

? MA9oeK@?7A9U= ;5n @HJ[; DFV eKM\4TV<=?XTJ[9o;>PFJBEM\@?@HMW=?4TZVU=?I16T@HJJ @T7:I6T@HJ< WPTJ[9UJp;>PFJDT4T= ; VUJ[Z\IJ[4L;MABN9<J 7A@l7@f=?V

Ω = [0, 1]=?VQXT=?V<eK9<J[;>=1UKJ[X ;>MI0 VU=H;UJKV_= x JAxE JDFVUJ

M = 10xe2mPT=?VQI1J 7A4TV

;>P87:;Y;UPTJ eKM\4L;>=H4SDTM\DTVQXTM\I7A=?4Ω=?VrVUD&5TV<;U=H;>Df;>JKX W=H;UP 7 VUJu;oMAB 6gM\=?4L;>V =H4 ;>PT=HVQeK7AVUJ ;>PFJ

VUJ[; 5gJK=H4TZ

Σ = 0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95.

4|M2;UPTJm6CJ[9U=HMfXT=HeO5gM\DT4TX87:9<nse[M\4TXT= ;>=HM\4TV M\DT@HXI1J 7A4s;UP87:;d;UPTJW6CM\=H4L;>V0.05

7:4TX0.95

VUJKJJ 7Ae PM:;>PTJ[9 7AVl4TJ[=?Z\P5gM\9U=H4TZ]6gM\=H4S;UVSJAx ZTxd@H=?tAJ

0.257A4TX

0.35x ?8DT9a;>PTJ[9UI1M\9UJ!i7A4Ln1BEDF4Te[;U=?M\4

hA7:@?DTJNeK7A41M\4T@Hn]P87 h\Jmh:7A@HDTJKVdM\4 ;UPTJWXT=?V<eK9UJu;>Jm6gM\=?4L;UVdMABgVUJ[;ΣL;>PFJNhA7:@?DTJ[V BEM\9 MA;UPTJK9 6gM\=?4L;>V

MABΩIsDTVa; 5gJr=?4L;UJK9U6gM\@?7:;>J[X)x

354 M\9UXFJK9;UM I7AtAJ;>PTJ`4SDTI1JK9<=?eK7A@W6T9UMieKJ[VUVpI1M\9UJ ;>9U7A4TVU6T7A9UJ[4S; J W=H@?@WVUPFM PTM ;>PFJXT=?V<eK9UJu;>=1U 7:;U=?M\41=?VlXTM\4TJw7A4TXXFJKI1M\4TV<;U9>7:;UJb;UPTJw4SDTI1JK9<=?eK7A@ IJu;>PTMiXTVl=?47Q; M)B XT=?I1JK4FVU=?MA487A@e 7AV<JAx 2mPTJ J@i;UJK4TV<=?M\4;>M;UPT9UJ[J1XF=?I1JK4TV<=?M\4TVw=?VrV<;U9>7A=HZ\PL;UBEM\9 N7:9UX7A4TX M\DT@HX =H4 ;>PT=HV e 7AV<JM\4T@HnwI7:tAJd;>PTJl4TMA;>7:;>=HM\4TV)I1M\9<Jle[M\I16T@?=He 7:;UJKXrXTDTJd;UM|P87 hi=?4FZW;UPT9UJ[JB XT=?I1JK4TV<=?M\4T7A@\I7:;>9<=?eKJ[VKx

zA

> =">B? K@T PWG K K@DAF

k 2ODT9U=H4TZoVanfVa;>J[I =?41; M XF=?I1JK4TV<=?M\4TV 6gM\VUJ[VNDTVm7Q6T9UM5T@?J[I WPTJ[9UJ J P87 h\J ;UMoXT=HVUe[9UJ[;U=6U[JJKRSD87:;U=?M\4TVWMABO;>PFJYBEMA@?@?MW=H4TZ]BEM\9PTJw75gM hAJwBEM\9<IsDT@ 7 ;>=?MA4s;>PTJb;>M\6TI1M\Va; J[RiDT7:;>=HM\4XTJuG84TJKV ;>PTJ|6T9UMK5F@?JKI/7A4FX;>PTJ|XTM\I7A=?4WPTJK9<J`P87AV;UM-5gJ`VUMA@Hh\J[X)x 2mPFJ 4TJ@i;; M JKRSD87 ;>=?MA4TVJ@f6T9<JKV<V;>PTJ6CJ[9U=?MiXT=He 5gM\DT4TXT7A9<neKM\4TXF=H;>=HM\4TV_7A4FX ;>PFJ@?7AV<;QJKRSD87 ;>=?MA4 G&@fJKVQ;>PTJ=?4T= ;>=?7A@_e[M\4TXT= ;>=?MA4TV W=H;>P PTJ[@?6 M:BNBEDT4Teu;>=?MA4g(x1, x2)

XTJ[G84FJKX =?41;>PTJ XFM\I7:=?4)x kwV 7AVNI1JK4L;>=HM\4TJ[X6T9UJuhi=?M\DTV<@Hn<F=?4;>PTJ|e 7AV<JrMAB2 DF9U=?4FZV<niV<;UJKI1V|;>PFJQBEDT4Fe[;>=HM\4

g=?VWDTVUDT7A@?@ n 7DT4F=HBEM\9<IXT=?Va;>9<=65TDF;U=?M\4 M:B 9>7:4TXTM\I 4SDTI 5gJK9<Vw9U7:;>PTJ[9

;>P87A47 BEDT4Teu;>=?MA4=H4;>PTJYDTV<D87A@ VUJ[4TVUJ:x

? MA9mVU=?I16T@H=?eK= ;5n ;>PTJY@ 7 ;U;>=HeKJre[M\4TV<;>7A4L; =dx = h

? =?Vme PFM\VUJ[4;>M 5gJQ=HVUMA;U9UM\6F=?e<f= x JAxEh = h1 =

h2

x 4|M ;UPTJ 4TM:; 7:;U=?M\4yij

=?V DFVUJKXpBEM\9 ;>PTJr7:6T6T9UM@f=?I7:;U=?M\4MABy(ih1, jh2, t)

f;>PFJwh:7A@?DTJwMABy7:;m;>PFJQ@?7:;U;U=?e[J VU= ;>J

(i, j)7:;W;>=HIJ

tx|V<=?4TZ ;>PF=?V JYZ\J[;W;UPTJYI7:;U9U= @

yh(t) =

y11 . . . y1Lyxxx x x x xxxyLx1 . . . yLxLy

.

4|J@i; J]XT=HVUe[9UJ[;U=6U[J ;>PTJrBEDT4Teu;>=HM\4fM hAJK9m;>PTJYXTMAI7A=H4pI7 ;>9U= @e5LnpV<J[;<;>=?4FZ

fh(t) =

f(h, h, t) . . . f(h, Lyh, t)xxx x x x xxxf(Lxh, h, t) . . . f(Lxh, Lyh, t)

.

7A4TXp;>PTJY=H4T=H;U= 7A@geKM\4FXT=H;U=?M\4TVW7:487A@?MAZ\M\DTV<@Hn

gh(t) =

g(h, h, t) . . . g(h, Lyh, t)xxx x x x xxxg(Lxh, h, t) . . . g(Lxh, Lyh, t)

.

|VU=H4TZp;>PTJ[VUJ1I7 ;>9U= @`4FMA; 7:;U=?M\4FV J14TM6F9UMieKJKJ[X 7:4TX 7:6T6T@Hn"VUM\I1J VUDF=H; 7)5T@?J I1J[;UPTMfX`;UMVUM\@ h\Jr;>PTJY687:9<;>=?7A@ XF=1XCJK9UJ[4L;>= 7:@)JKRSD87:;U=?M\4 x

z.0

> ="> = 4JLM@G PT RG LD

2mPTJylDF@?JK9lW V I1J[;UPTMfX =HV J[@?@ B5tS4TMW4 e[=H;UJK6TP87:9U9U=HVp7A4TX V<=?I16T@?J"BEMA9pVUM\@ hf=H4TZ XT=YX J[9UJK4L;U= 7A@JKRSD87:;U=?M\4TV[x%2mPTJ I1J[;UPTMfX"=?V 5T7AVUJ[X M\4 4SDTIJ[9U=He 7A@O=?4L;>J[Z\9>7 ;>=?MA4MAB XF=1XCJK9UJ[4L;>= 7:@ J[RiDT7:;>=HM\4TVKx354`M\DT9be 7AV<J< JoDFVUJs= ;|BEM\9W;>PFJo=?4L;UJKZ\9U7:;>=HM\4M hAJK9w;UPTJ];U=?I1JQVa;>J[6)x%?_=H9UVa; JoZ\=HhAJo7V<PTM\9a;9UJ[hi=HJ (M:B ;UPTJYIJu;>PTMiX =H;>V<JK@ B x

354 ;UPTJrZ\J[4TJK9U7A@)e 7:VUJ<8y DT@HJK9lW VmI1J[;UPTMfXp=HVNDTV<JKX BEM\9NVUMA@Hhi=?4TZ V<=?I16T@?JbG89UVa;WM\9UXFJK9mXT=YX J[9UJK4L;U= 7A@JKRSD87:;U=?M\4 MABw;UPTJBEM\9= h\J

y′=HV7:6T6T9UM@f=?I7:;UJKX 5Sn

XT=1XCJK9<JK4Te[JQRSDTM:;>=?J[4L; JQM5F; 7A=H4p;>PTJYJKRSD87 ;>=?MA4

y(ti + δ) − y(ti)

δ≈ f(ti, y(ti)) + O(δ),

= Fx ?

WPTJK9<JO(δ)

=?VW7:4J[9U9UMA9N;>JK9<IMAB M\9UXFJK9δTVUDTe P;>P87 ;

limδ→0

O(δ)

δ= 0.

= npXT9<M\6T6T=H4TZ ;>PTJYJK9<9UM\9N;UJK9PTJ[9b9<JKXTDFeKJKX;>M

y(ti + δ) ≈ y(ti) + f(ti, y(ti)) · δ.= Fx93 ?

354 M\DT9]XT=?V<eK9UJu;>Je2ODT9<=?4TZ`VaniV<;>J[I JG&@ ;>PTJ1;>=HIJ1V<;UJK6 7AVδ= MA9

dt?>x = n VUJu;U;>=H4TZ

ykh =

yh(kδ) JQM5F; 7A=H4pBEM\9my DT@?J[9'W VbI1J[;UPTMiX

yk+1h = yk

h + δ(D∆hykh + f k

h )

y0h = gh,

= fx<7 ?

WPTJK9<J∆h

XTJ[4TMA;UJKVd;>PTJ XT=?V<eK9UJu;>=1UKJKX @?7A6T@?7AeK=?7A4 ;>PTJNXFJK9U= h:7:;>=HM\4sMAB WPT=He P W=?@?@V5gJN6T9UJ[VUJ[4S;UJKX4TJ@i; x

> ="> KBFK QG LK G PQGIF UG RG LD

2mPTJ XF=1XCJK9UJ[4TeKJlI1J[;>PFMfXQ=?V W=?XFJK@Hn DTVUJ[Xs=?4Q6T9UM5T@?J[IV =?4LhAM\@Hhi=?4FZW;UPTJl4iDFIJ[9U=?eK7A@Le 7:@?eKDF@ 7:;U=?M\4MAB ;UPTJ1@ 7A6T@?7AeK=?7A4`M\6gJK9U7:;>MA9 ∇2 = ∆

x1354 M\DT9r; M)B5XF=?I1JK4TV<=?M\487:@_V<;UDTXFn< ;>PFJ1@?7A6T@?7AeK=?7A4 MAB;>PTJrBEDT4Teu;>=HM\4

y(x1, x2)e 7:4 5gJ W9U= ;U;>J[4MADF;b7AVmBEM\@?@HMWV

∆y =∂2y

∂x21

+∂2y

∂x22

.= fxb8 ?

z\z

354;UPTJQ;UPT9UJ[JB XT=?I1JK4TV<=?M\4T7A@ eK7AVUJ!g;UPTJYM\6CJ[9>7:;UM\9 M\DT@?XeKMA4S;>7A=?4VUJ[eKM\4FX"XTJ[9U=Hh:7:;U=HhAJ]7:@?VUM =H4;>PTJo;UPT=?9<X XT=?9<JKeu;>=?MA4)x = n"7A6T6T9<M@f=HI7:;U=?4TZ1;>PTJ XFJK9U= h:7:;>= h\JKV|; W=?eKJo=?4`;>PFJsV>7AI1J 7 n7AV =H4;>PTJY6T9<J[hi=?M\DFVbV<JKeu;>=?MA47:4TXDFVU=?4FZ ;UPTJY4TMA;>7:;>=HM\4

yi,j

BEM\9y(hi, hj)

JQMK5F; 7:=?4

∂2y

∂x21

≈ yi+1,j − 2yi,j + yi−1,j

h2.

= fx<; ?

= n 9UJ[6CJK7:;>=H4TZr;>PTJ|75gM h\JW6T9UMieKJ[XTDT9<JbBEM\9dXFJK9U= h:7:;>= h\JKV W= ;>P19UJKV<6CJ[e[;l;>My Jbe 7:4 W9<=H;>Jm;UPTJ

XT=?V<eK9UJu;>=1UKJKX@ 7:6T@ 7Ae[= 7A4M\6gJK9>7 ;>M\9NBEM\9my R x = Fxb7 ? 7AV∆h = I ⊗ ∆h1

+ ∆h2⊗ I

WPTJK9<J

∆hp=

1

h2p

−2 1 . . . 0 1

1 −2 . . . 0 0xxx x x x x x x x x x xxx0 . . . 1 −2 1

1 . . . 0 1 −2

.

BEM\9p = 1, 2

x 4bMA;>=HeKJ";UP87:; 6gJK9<=?MiXT=?e 5CM\DF4TX87A9an eKM\4FXT=H;U=?M\4TV7A9UJ 6F9UM\6gJK9<@Hn2;>7AtAJ[4 =H4L;>M7AeKe[M\DT4L; x

& 6 + (*"+ :I+ /214. 6 + ,

? MADT9U=HJK9d;>9U7A4TV<BEMA9UI=?V 7 W=HXTJK@ n DTVUJ[XIJu;>PTMiXBEMA9 V<6CJ[e[;>9U7A@C7A487A@ nfV<=?V 7A4TX7A4=?4L;U9UMiXTDTe[;U=?M\4;>M";UPTJVUD&5 <JKeu; e 7A4 5gJBEM\DT4FX BE9UM\I 7A4LnI587AVU=He:5gMfMAt PT7A4TXT@H=?4TZDT4T=HhAJK9<VU=H;5n @?Juh\J[@NI7 ;>P.BJKI7:;>=HeKVKx 2mPTJs487:IJ]=?V|;UMPTM\4FM\9 7 ?89UJ[4Te P VUe[=?J[4S;U=?Va; \JK7A4.B =N7A6F;U=?V<;UJ\M\V<JK6TP ?8M\DF9U=?J[9C7eKM\4L;>J[I6gM\9U7A9<npM:B[4|7A6CMA@?JKMA4 WPTM1=?4Lh\J[4L;>JKX;>PTJQ=?XFJ 7 MAB XTJKV<eK9U=15T=?4FZ7sBEDT4Teu;>=HM\4"7:V|7 @H=?4.BJ 7A9WeKMAI 5T=?4T7:;>=HM\4MAB_MA;>PTJ[9NBEDT4Teu;>=?MA4TVKxlkwVm;>PFJZ?8M\DF9U=?J[9mVUJ[9U=?J[VW=?VmJ@i;UJK4TXTJ[X";>M e[M\4L;>=?4SDTMADTVe 7AV<J;>PTJV<DTI 5gJKe[M\I1JKV 7A4 =H4S;UJKZ\9U7A@ 7A4TX Jpe 7A4 XTJuG84TJp;>PFJM\4TJ B5XT=HIJ[4TVU=HM\487A@ ?8M\DT9<=?JK9;>9>7:4TV<BEM\9PTJrBEDT4Fe[;>=HM\4

h(t)7AV

H(ω) =

∫

∞

−∞

h(t)eiωtdt,= FxbS ?

7A4TXp;>PTJY=H4ShAJK9<VUJY;>9>7:4TV<BEM\9PFJo;>=HIJo7A4TX

ω;>PTJs7A4TZ\DT@?7A9|BE9<JKRSDTJ[4Te[nAx |J[9UJ< J 7A4L;r;>MMK5f; 7A=H4 7

;>PT9<JKJB XT=?I1JK4FVU=?MA487A@f;U9>7A4FV<BEM\9<IM\4T@ n W=H;UP9<JKVU6gJKeu; ;UMYVU687Ae[JAx 2mPT=HV e[M\DT@HX 5gJm;>PTMADTZ\PL;lMAB)=H4

zA

;>PTJrBEM\@H@?MW=?4TZ N7 ngjtP87AVm;>PFJQDF4T=H;WV<JKeKMA4TX

s7A4TX

ω;UPTJYDT4T= ;

1/sx[] PTJ[4 JQ;U9>7A4FV<BEM\9J~xWPT=He P P87AVdXTJ[VUeK9<=65gJKV @?J[4TZA;UP\JAx ZTx DT4T=H;UV

m JbZAJ[;dV<M\IJu;>PT=H4TZ W=H;UP

DT4T=H;1/mx[2mPT=HVd=?VdMAB eKM\DF9UVUJm;UPTJ N7 h\Jbh\J[e[;>MA9 ~k x\] PTJK4 J W9<=H;>Jm;UPTJ ? MADT9U=HJK9_;U9>7A4TVaBEM\9UIBEM\9mM\DT9m6TDF9U6gM\VUJ[V J]ZAJ[;

H(~k) =

∫

Ω

h(~x)ei~k·~xd~x,= fx10 ?

WPTJK9<JΩXFJK4TMA;UJKV_;UPTJ ;UPT9UJ[JB5XF=?I1JK4TV<=?M\487:@fXTMAI7A=H4o7:4TXs;>PTJ M\6CJ[9>7:;UM\9 · ;>PTJ VUe 7:@ 7A9 6T9UMiXTDTeu;MAB ;UPTJr; M h\J[e[;UM\9UV[x

> >B? K@T PWG G D JPWK@G P P FLT D P R

354M\DT9 6T9<MK5T@HJKI S;UPTJbX87:; 7Y=?V XT=HVUe[9UJ[;UJ 7A4FX=?4FV<;>JK7AX1MAB);>PFJ|e[M\4L;>=?4SDTMADTVl;U9>7A4FV<BEM\9<IXTJ[G84FJKX5Sn y R x2= FxY0 ? JwP87 h\Jb;>MQDTVUJbXT=HVUeK9<J[;UJ%?8M\DT9<=?JK9 ;U9>7A4FV<BEM\9xSk 5T9<=?JuB)=?4L;U9UMiXTDTe[;U=?M\4;>Mv%?\2e 7A4^5CJYBEM\DT4TXBE9UM\ID 9UJKV<V J[; 7A@ xE0TKTK76 xf2mPFJs=?XTJK7=HVb;UM;>9U7A4TVaBEM\9UI;UPTJo=H4S;UJKZ)B9>7:;U=?M\4 ;>M7VUDFI M:BdXT=?V<eK9<J[;>JQh:7A@?DTJ[VwV<DTe P";>PT7:;w;UPTJ]v%?\2eKM\9<9UJ[VU6gM\4TXT=H4TZ;UMy R x = fxbS ? =?VZ\=HhAJK4H5Sn

H(ωn) = ∆

N−1∑

j=0

hjeijn/N ,

= FxY0K0!?

= x JAxECXT=HVUe[9UJ[;UJs6gM\=?4L;>Vhj

MABd;UPTJ];U=?I1J]XFM\I7:=?47A9UJo;U9>7A4FV<BEM\9<IJ[X;>MXT=?V<eK9UJu;>Js6gM\=H4S;UVωn

MAB;>PTJpBE9UJ[RSDTJK4Teun XTM\I7A=?4 x 2mPTJ e 7A@HeKDT@?7:;>=HM\4 MAB|v ?\2 =?V 4TM:;Va;>9U7A=?Z\PL;<BEM\9 7A9UX XFDTJ;UM=H;>V;>=?I1Jwe[M\4TV<DTI=H4TZ 487:;UDT9UJ:x 2mPTJK9<JY7A9UJrV<J[h\J[9>7A@O7A@HZ\M\9<=H;>PFIV BEM\9NeK7A@?e[DT@ 7:;U=?4TZQ;>PTJ ;U9>7A4FV<BEM\9PTJYI1M\V<;WDTV<JKXH5CJ[=?4TZ ;UPTJrB 7AV<; ?8M\DT9<=?JK9 ;>9>7:4TV<BEM\9<IA= ?[?\2)? x

> > =

3 ;d=?VdDTV<JK@HJKVUV ;UM 5TDT=?@HX 7 eKMiXTJNBEM\9[? MADT9U=HJK9 ;U9>7A4FV<BEM\9PTJmV<=?IsDT@?7:;>=HM\4oe[MfXFJf5CJ[e 7ADTV<J;>PTJ[9UJ 7A9<J I7A4Ln"@?=65F9>7A9<=?JKV|XTJ[hAJK@HM\6CJ[X BEMA9w;>PT=HV ; 7AV<t x ? 7AV<;UJKV<;Z?8M\DT9<=?J[9 2O9>7A4TVaBEM\9UI =?4`;>PFJ] J[V<;4= ?[?\2O] ?% ?89U=HZ\M7A4FX \M\PT4TV<M\4 e[=H;UJKX 7fxY0 zfx zA\.0O=?V 7BE9<JKJsV<MAB ; 7A9UJsXFJ[h\J[@?M\6gJKX7 ;;>PTJ ^`7:VUV>7:e PiDFVUJ[;<;>V354FV<;>= ;>DF;UJpMAB 2 J[e PT4TM\@HM\ZAn\x 3 ;=HVsJ 7:V<n ;>M DFVUJ7A4TX MXCJK9UV 5gMA;UP 9<J 7A@7A4TX e[M\I6F@?J@ ;U9>7A4TVaBEM\9UI1VQ=?4 M\4FJM\9]IM\9<JXF=?I1JK4TV<=?M\4TV[x 4TJMABN;>PFJZ\9UJK7:;oBEJK7:;>DF9UJKVoMAB;>PTJ`@H=65T9U7A9U=HJKV =?V1;>P87 ;;>PFJ[n27AX87A6F;;>M I7A4Ln 6T@ 7:;<BEM\9UI1V17A4TX eK7A4 M\6F;U=?I1=6UKJp;UPTJKI1VUJ[@Hh\J[V7ADF;>MA4TM\I1M\DTVU@ n\x 3547AXTXF=H;>=HM\4i;>PTJr;>9U7A4TVaBEM\9UI =?VNh\J[9<npB 7AVa;|7AVN;>PTJY487:IJrVa; 7:;UJKVKx

z)3

& 6 1 * 4, /2141 . :</ (: /714/ . .

2mPTJlhAM\4Z4|JKDFI7A4F4]Va; 75T=H@?= ;5nr7A487:@HniVU=HV=?V 7WIJu;>PTMiXrBEM\9O7A487A@ n.U[=?4TZmVa; 75F=?@?= ;5nwM:B87WXT=1XCJK9<JK4Te[JJKRSD87:;U=?M\4)x k 9UJuhi=?J MAB ;UPTJNI1J[;>PFMfXseK7A4 5CJ BEM\DT4TXsBEM\9 J@F7AI16T@?J BE9UM\I l9<JKV<VlJu; 7A@ xBV0TKT(7 xkw6T6T@H=?eK7:;>=HM\4MABO;>PTJYI1J[;UPTMfX M\4p;UPTJQ75CM hAJ]6T9<MK5T@HJKI W=H@?@j5CJY6T9<JKVUJ[4L;>JKX"4TJ@i; x

2mPTJYXT=1XCJK9<JK4Te[JQJ[RSD87:;>=HM\4=?4p;>PT9<JKJYXT=HIJ[4TVU=HM\4TVNJ[RiDT7:;>=HM\4=HVmZ\=HhAJK4 5Ln

yt+1mnp − yt

mnp

δ=

D

h2(ym+1,n,p + ym−1,n,p + ym,n+1,p + ym,n−1,p + ym,n,p+1 +

+ ym,n,p−1 − 6ymnp) + f tmnp(y),

= fx10 z0?

WPTJK9<JhXTJK4TM:;>JKVW;>PFJQ@?7:;U;U=?e[J eKM\4FV<; 7:4S;|7A4FX

δ;UPTJr;>=HIJrVa;>JK6 x

Ju;sDTV]4TM eKM\4FVU=?XFJK9];UP87:;Q;>PTJXT=1XCJK9<JK4Te[JJKRSD87 ;>=?MA4 h:7A9U=HJKVQVUM`V<@?MW@Hn ;UP87:; Je 7A4 ;U9<n=?4TXTJ[6CJ[4TXTJ[4S;|VUMA@?DF;U=?M\4TV MAB ;UPTJrBEM\9UI

ytm,n,p = ξteikxmheikynheikzph,

= FxY0 ?

WPTJK9<Ji=HVQ;UPTJ=HI7AZA=?487A9an`DT4F=H; _7A4TX

kx

ky

7:4TXkz

;>PFJ 7 hAJ4SDTI 5gJK9<Vs=?4 @VB n(BY7:4TXUB5XF=?9UJ[e[;U=?M\4TVf9<JKV<6CJ[e[;>= h\J[@Hn\x 3 ;mV<PTM\DT@HXe5gJ 4TMA;UJKX;>PT7:;N;>PFJ @ 7:;<;>=HeKJbeKM\4TVa; 7A4L;

h=?Vl;>PTJ VU7AI1J

=?47A@?@CXT=?9<JKeu;>=?MA4TVW7AVWXT=?V<eKDTV<VUJ[X^5gJ[BEM\9<JAx

] PTJK4 JQVUD&5TV<;U=H;>Df;>JYy Rgx8= FxY0 ? ;>M y R x = FxY0 z ? 7A4FXV<=?I16T@?= B n JQZ\J[;

ξ =Dδ

h2(eikxh + e−ikxh + eikyh + e−ikyh + eikzh + e−ikzh − 6) + 1 +

δf th(y)

ξn(emkxenkyepkz)ih,

= FxY03?] Jle 7A4QBEDT9<;UPTJK9 7A6T6F9UM@f=?I7:;>J

ξ(k)5Lnr;>9>7:4TV<BEM\9<I=H4TZWeKM\I16T@HJ@Q4SDTI 5gJK9<VO;UM|;>9<=?Z\MA4TM\I1J[;>9<=?e

BEDT4Te[;U=?M\4FVwDTV<=?4TZy DT@HJK9lW VwBEMA9UIsDT@?77:4TX 5Ln"I1=?4FM\9U=H4TZ1;UPTJse[M\VU=H4TJKV W= ;>PI=H4iDFVwM\4TJ:x%2mPT=HVnf=HJK@HXTV

|ξ(k)| ≤ |1 − 12Dδ

h2+

δf th(y)

ξn(emkxenkyepkz)ih|. = FxY07 ?

] J7A9UJ=?4L;>J[9UJKVa;>J[X =?4 ;UPTJI1MfXFDT@?DTVQMABξ_e 7:@?@?J[X ;UPTJ*< ,@ C$!1 &3 $!+ux a MA4TXT=H;U=?M\4

|ξ(k)| < 1BEM\9Q7:@?@

k =HI6T@H=?J[Vw;>P87 ;Q;UPTJ XT=1XCJK9<JK4Te[JJ[RSD87:;>=HM\4=?VYV<;>75T@HJ< = x JAxE 4TMI1MiXTJ =?V

J@f6CMA4TJK4L;>=?7A@?@ nsZ\9<MW=?4TZTx2mPTJb@ 7AVa;l;UJK9<IM\41;UPTJW9U=HZ\PL; V<=?XTJWMAB y Rgx@= Fx107 ?O=?V XTJ[6CJ[4TXTJK4L; M\4;>PTJm9<J 7Aeu;>=?MA4stS=?4TJu;>=?e[VdMAB ;>PTJmJ[RSD87:;>=HM\4)x2mPiDFV JmP87A4FXT@?J ;UPTJmZ\J[4TJK9U7A@M2ODT9U=H4TZrV<niV<;UJKI7:4TX;>PTJ_r9>7 n(Bacfe[MA;U;NIMiXTJK@CVUJ[687A9>7 ;>JK@ n\x 4bMA;>=HeKJ ;>PT7:;m=?4 7:XTXT=H;U=?M\4;>PTJ ; M JKRSD87:;U=?M\4TVmM:B JK7Ae PV<niV<;UJKI-IsDFV<; 5gJ 7A487A@ n.U[JKXVUJ[687A9U7:;>J[@Hn5gJKeK7ADTV<J ;UPTJ V<;>75T=H@?=H;5ne[M\4TXT= ;>=?MA4TV 7A9<J XT=1XCJK9<JK4L; x|M Juh\J[9

δ7:4TX

hIoDTV<; 5gJse PTMAVUJK4J[RiDT7A@ BEM\9 5CM:;>P JKRSD87 ;>=?MA4TV MABl7VaniV<;>J[I-XTDTJ];>M;>PFJ

zK7

eKM\DT6F@?=?4FZ MAB ;UPTJYJKRSD87:;U=?M\4FVKx

> >B? GIFLG P M J PWK@F T 9T G R

3 ; W=?@H@ 5gJpVUPTMW4 =?4 a PT7A6F;>J[9 3`;>PT7:;s;>PFJpti=H4TJ[;U=?e[VsMABW;>PTJpZAJK4TJ[9>7A@ 2 DT9<=?4TZ VaniV<;>J[I 7A9<JZ\=HhAJK47AV

f(u, v) = αu(1 − r1v2) + v(1 − r2u)

g(u, v) = v(β + αr1uv) + u(r2v − α).= fx108 ?

kwV J @H=?4TJK7A9U=1UKJm;>PTJw75gM hAJ JKRSD87 ;>=?MA4TV Jr7:9UJb@?J[B ; W=H;UPf(u, v) = αu+ v

7A4TXg(u, v) =

βv − αux 2OMI7At:JQ;UPTJQ7A6T6T9<M@f=HI7:;U=?M\4p9<J 7AV<M\4875F@?J J]DTV<JQ;UPTJQV>7AI1Jr;>9U=?7A@ V<M\@?Df;>=?MA4 BEMA9

5CMA;UPu7A4TX

v=?4 ;>PTJpBEM\9P87 ; J I7:tAJ7 6TP87AV<JV<PT=HB ; MAB

π9>7AXT=?7A4TV 5CJu; J[JK4u7A4FX

vx#2mPT=?Vb=?VbXTDTJQ;>M;UPTJ]e[9UM\V<V B57Ae[;U=Hh:7:;>MA9 B5=H4TPT=15T=H;UM\9N487:;UDT9UJ]MAB ;UPTJ

V<niV<;UJKIx

] PTJK4 JbVUDM5TVa;>= ;>DF;UJwM\DT9l@?=H4TJ 7:9U=6U[JKX tS=?4TJu;>=HeKVd;>M]ylR x@= fx1070?d7A4FXV<=?I16T@?= B no;>PFJ|J[RiDT7:;>=HM\4TV JYZ\Ju;

max |ξu(k)| = |1 − δu(12Du

h2− 1 − α)|

max |ξv(k)| = |1 − δv(12Dv

h2− β + α)|. = Fx10; ?

?_=?Z\DF9UJYFxY0]V<PTMWV|;UPTJY6T@?MA;UVWMAB y R x = FxY0; ? BEM\9max |ξ(δ)| WPTJK4;>PTJQ@ 7 ;U;>=HeKJre[M\4TV<;>7A4L;w=HVG&@fJKX;UM

h = 17A4FX687A9U7AI1J[;>J[9dhA7:@?DTJ[V e[M\9U9<JKVU6gM\4TX;>MQXT=1XCJK9<JK4L; I1MiXTJKV[x[2mPTJwV<JK@HJKe[;U=?M\4MAB

;>PTJ|687A9>7:IJu;>JK9 h:7A@?DFJKV W=?@?@5gJwXF=?VUe[DTVUV<JKXp=H4I1M\9UJbXTJu; 7A=H@g=H4ea PT7A6F;>J[9UVO3o7A4TX 7ix ?8M\9 4TM%;>PTJ]687A9>7:IJu;>JK9bh:7A@?DTJ[VwV<PTM\DT@HX 5CJ]eKM\4FVU=?XFJK9UJ[X <DTV<; Z\=HhAJK4)x 2mPTJsVUDF=H; 7)5T@?JYh:7A@?DTJ[VwMAB

δ7A9<J

;>PTJsMA4TJKV WPFJK9UJmax |ξ(δ)| =?V 5CJ[@?M;>PFJsDT4T= ;5nWPF=?e P P87AV 5CJ[JK4 XT9U7W4 =H4 ?_=HZ\DT9UJ[V FxY07A4TX fx z BEM\9e[@ 7A9<=H;5nAx 4|MA;U=?eKJ ;>PT7:;;UPTJ^5gMA;U;UM\I 9UM M:B ?_=?ZADT9UJFxY0"=?I16T@H=?JKV 6F9UMK5T@HJKI1V

5CJ[e 7ADTV<J W= ;>P ;>PFJ ;UPT=?9<X VUJu;MAB|687A9U7AI1J[;>J[9UVs;UPTJpG89UVa;J[RSD87:;>=HM\4 BEM\9 e PTJ[I=He 7A@u=?V 4TM:;

V<; 7)5T@?J BEM\9Y7A4Lnδx 2mPF=?VYe 7A4 5gJ1V<JKJ[4 BE9UM\I ?_=?ZADT9UJ FxY0 J VU=H4TeKJ ;>PFJ7:I6T@H=HG8eK7:;>=HM\4"B 7Aeu;>M\9

=?Vo@?7A9UZAJK9Y;>P87A4 M\4TJ7:4TX ;>PSDTV DF4TV<;>75T@?J1BEM\9s7:@?@dh:7A@?DTJ[VoMABδx bM Juh\JK9 =?4 6T9U7Ae[;U=?V<J;>PFJ

4TM\4T@H=?4TJK7A9 JX J[e[;>VW7:4TXp;>PTJ e[M\DT6T@H=?4TZoMAB ;>PTJrJ[RiDT7:;>=HM\4TVNVa; 75T=H@?=1UKJW;>PTJr= ;>JK9U7:;>=HM\4)x\=N7AVUJ[X M\4;>PT=HV|7:487A@HniV<=?V J]BEM\DT4FX;UP87:;Wh:7A@?DTJ[V

h = 17A4FX

δ = 0.05 JK9UJ]VUDT= ; 7)5T@?J BEM\9m;UPTJ]ZAJK4TJ[9>7A@

2 DT9<=?4TZ VanfVa;>J[Ix

z8

0 0.2 0.40

0.5

1

1.5

δu

max

|ξ(k

)|

0 0.2 0.40

0.5

1

1.5

δv

max

|ξ(k

)|

0 0.5 1 1.50

0.5

1

1.5

δu

max

|ξ(k

)|

0 0.5 1 1.50

0.5

1

1.5

δv

max

|ξ(k

)|

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

δu

max

|ξ(k

)|

0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

δv

max

|ξ(k

)|

?_=?Z\DF9UJ]Fx10Aj l@HMA;>V|MAB y R x = fx10;0? BEM\9b;>PTJoDTVUJ[X VUJ[;UVrMABd687:9>7AI1J[;UJK9UV[x ?_=HZ\DT9UJ[V|MA4";>PTJo@?J[B ;eKM\@HDTI4 = 7 Se|7:4TXJ ?_e[M\9U9<JKV<6CM\4FX;>MY;>PFJbJ[RSD87:;>=HM\4TVlMAB

u7A4TX 5SX7A4FX1BC;>MY;>PTJbJKRSD87:;U=?M\4FV

MABvx 2mPTJ687A9U7AIJu;>J[9UV MABb;>PTJV<niV<;UJKI JK9UJ=?4 77A4TX 52?

Du = 1.032Dv = 0.516

α = 0.899

7A4TXβ = −0.91

be"7A4TX X@?Du = 0.244

Dv = 0.122

α = 0.398

7:4TXβ = −0.4

TJQ7A4FXB"?Du = 0.129

Dv = 0.516

α = 0.89

7A4TXβ = −0.99

x

zK;

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

δu

max

|ξ(k

)|

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

δv

max

|ξ(k

)|

?_=?Z\DF9UJ1Fx zfj> @?MA;UV]M:Bmy Rgx = FxY0T ?bBEM\9Y;>PFJDTV<JKX VUJu;sMABN687A9>7:IJu;>JK9<VKxH?_=HZ\DT9UJ1M\4 ;>PTJ@?J[B ;eKM\9<9UJKV<6CMA4TXTV ;UM];>PFJwJKRSD87 ;>=?MA4 W=H;UP

u7A4FX;>PTJ|M\4TJ|M\4;UPTJw9<=?Z\PL;l;>M];UPTJwJ[RSD87:;>=HM\4 W= ;>P

vx

2mPTJY687A9>7:IJu;>JK9<V J[9UJDu = 0.125

Dv = 0.05

F = 0.045

7A4FXK = 0.065

x

> > = P UD RD GIM

3 ; W=H@?@5gJ|V<PTMW4=?4:a P87A6F;UJK9O7Y;>PT7:; ;UPTJbti=H4TJ[;U=?e[V MABC;>PTJ%_Y9U7 nB cfeKM:;U;lIMiXTJ[@C7:9UJbZ\=HhAJK47:V

f(u, v) = −uv2 + F (1 − u)

g(u, v) = uv2 − (F + K)v.= Fx10S ?

] PTJK4 ;UPTJJ[RiDT7:;>=HM\4TV 7A9<J@H=?4TJK7A9U=1UKJKX 7A4TX ;>PTJe[M\4TVa; 7A4L;s;UJK9UI=HVo4TJ[Z\@?J[e[;>J[X J7A9UJ@?J[B ;W=H;>P

f(u, v) = −Fu7A4TX

g(u, v) = −(F + K)vx = nBEMA@?@?MW=H4TZ];UPTJrV>7AI1J 6T9<Mfe[JKXTDF9UJQ7AV

=?4p;>PTJY6T9<J[hi=?MADTVWVUJKeu;>=HM\4 J]ZAJ[;

max |ξu(k)| = |1 − δu(12Du

h2+ F )|

max |ξv(k)| = |1 − δv(12Dv

h2+ F + K)|. = Fx10T ?

?_=?Z\DF9UJlFx zWV<PTMWV ;>PFJ 9UJ[VUDT@ ;>=H4TZW6T@?MA;UVOBEMA9OMA4TJ M:BF;>PFJ DTV<JKXsI1MfXFJKVKx=N7AV<JKXsMA4Q;UPT=?V 7A487A@ niVU=?V7A4TX4SDTI1JK9<=?e 7:@g;UJKV<;UV Jre PFM\VUJ h:7A@HDTJKV

h = 17:4TX

δ = 1x 4|MA;U=?e[J|;>PT7:; ;UPT=?V IMiXTJ[@ 7A@H@?MWV

IsDTe PH5T=?Z\ZAJK9N;>=HIJwV<;>J[6;UP87A4 ;>PTJYZ\J[4TJK9U7A@2ODT9U=H4TZ IMiXTJK@ x

& 6 , 8 :</: 6714/ 3;*":</ .

kwV M\4TJ:5CJ[Z\=?4TV];>M W9U= ;>J7`VU=?IoDT@ 7:;U=?M\4 6T9UMAZ\9>7AI _;UPTJBEM\@H@?MW=?4TZRSDTJKVa;>=?MA4TV =?4 7:XTXT=H;U=?M\4;>M ;>PTJ 4iDFIJ[9U=?eK7A@Y7A@HZ\M\9U= ;>PTI VUJ[@?J[e[;>=HM\4(7:4TX7:487A@HniV<=?VVUPTM\DF@?X 5CJ eKM\4FVU=?XFJK9UJ[X)j] PT=He P

zS

6T9UM\ZA9>7AI1I=H4TZ@ 7A4TZAD87AZ\JV<PTM\DT@HX 5CJpDTV<JKX |M/;>PTJpe[MfXTJpeK7A4 5CJpMA6F;>=HI=1UKJKX-5Sn 6T9UM)BZ\9>7AI1I1=?4TZ ;>9<=?e tSVY7A4TXDTV<=?4TZpe[M\I6F=?@?J[9 7 7:Z\V |MVUPTMADT@?X ;>PFJ X87:; 7e5gJ 9UJK7AX`BE9UMAI%;>PFJ6T9UM\ZA9>7AI |M eK7A4 ;UPTJ9<JKV<DT@H;UV 5gJ hf=HVUD87:@?=6U[JKX -354 ;>PT=HV1V<JKeu;>=?MA4 J"e[M\4TeKJ[4L;>9>7 ;>JM\4;>PTJ G89<V<;W;>PF9UJKJYRSDTJ[V<;>=HM\4TVb7A4TXVUMAIJrMA;UPTJK9N;UJKe PT4F=?e 7:@XTJ[;>7A=?@HVKx[2mPTJY@?7AV<;mRSDTJ[V<;U=?M\4 W=?@H@5gJXT=?V<eKDTV<VUJKX=?4p;>PFJQ4FJ@i;bVUJKeu;>=HM\4)x

2mPTJ`VU=HIsDT@?7:;>=HM\4 6T9UM\ZA9>7AI 7AVI7AXTJ"DFVU=?4FZ a6T9UMAZ\9>7AI1I1=?4TZ@ 7:4TZ\D87AZAJ<NV<=?4Te[J`=H;=?V7VU=?I16T@HJ%5TDF;Wh\J[9UVU7:;>=H@?Jr@ 7:4TZ\D87AZAJY7A4TX=H;m=HVbtS4TMW4;>P87:; a ;>M\ZAJ[;>PFJK9 W=H;UP ?8M\9a;>9U7A47A9UJr;>PFJIM\Va;YJ1eK=HJK4L;Q@ 7A4FZ\D87AZ\J[V BEM\9YeKM\I16TDF;>7:;>=HM\487A@O; 7AV<tSVKx1kw4 7AXTXT= ;>=?MA487A@ 7AXFh:7A4L; 7AZAJ1=HV ;>P87 ;^"k 2 k%= MX J[9UVY7 W=HXTJ 7A6T6T@H=?eK7:;>=HM\4"6T9<M\Z\9U7AI =H4S;UJK9aB 7AeKJoBEMA9 5CMA;UP MABl;>PTJ[VUJ @ 7A4FZ\D87AZ\J[VKx2mPTJo=H4S;UJK9aB 7AeKJQe 7A4 5CJQDTV<JKX`;>M1J@fe P87:4TZ\JsX87 ; 7 5gJ[; JKJ[4^"k 2 k%= 7A4TX";UPTJ]V<=?IsDF@ 7:;U=?M\46T9UM\ZA9>7AI M\9N;>M DTV<Jo^"k 2 k = W VWBEDT4Teu;>=HM\4TVm=?4p;UPTJY6T9UM\ZA9>7AI MA9Whi=HeKJ h\J[9UVU7Fx

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2mPTJ]M:;>PTJ[9Whi=HVUD87A@H=6UK7:;>=HM\4VUM:B ; N7A9<JQ;UP87:; 7AV|DFVUJKX; 7AV|7A4MA6CJ[4V<M\DT9Ue[J]V<MAB ; 7A9UJQe 7A@H@?J[X 6gJK4Fv XTJuh\J[@?M\6gJKX 5Ln 3C=N^ x 6gJK4Tv JK487)5T@?J[V ;UPTJm9UJK7A@YB ;>=HIJ hi=?VUDT7A@?=1U 7:;U=?M\4sM:Bg;UPTJWVU=HI BDT@ 7:;U=?M\4 DFVU=?4FZs7YVUM\6TPF=?V<;U=?eK7:;>J[XX87 ; 7Q;U9>7A4FV<BEJK9lI1J[;>PFMfXTV 5gJ[; JKJ[4; M]eKM\I16TDF;UJK9UVf= x J:xB JVU=?IoDT@ 7:;UJ ;>PTJQV<niVa;>JKI DTV<=?4TZ7 B 7:V<;|6T9UMieKJ[VUVUMA9|M\9WI7A4Ln 6T9UMieKJ[VUV<M\9UVw7A4TX;UPTJQX87:; 7 =?VWV<JK4L;M\DF;7A@?I1M\Va; eKM\4L;U=?4SDTM\DTV<@Hn ;UM ;UPTJphf=HVUD87:@?=6UK7:;>=HM\4 6T9<M\Z\9>7:I WPT=He P2XTMiJKV14TMA;1P87 h\J ;>M 5gJ9>7A4 M\4;>PTJ1V>7AI1J eKM\I16TDF;UJK9Q7AVr;>PFJV<=?IsDT@?7:;>=HM\4)xs354;>PF=?V N7 n<_=H;Y=HVYV<;>9U7A=?ZAPS;<BEM\9 7A9<X;UMI7At:JwI1M hf=HJKV VUPTMW=?4FZo;>PTJ|XTJ[hAJK@HM\6TI1JK4L;mMAB;UPTJwI1M\9U6FPTM\@?MAZAn\x\? MA9 XT=HVUe[DTVUV<=?M\4p7)5CM\Df; ;>PFJ7AXFh:7A4L; 7AZAJKVm7A4FXXT=?VU7AXFh:7A4L; 7:Z\JKVNM:B 6gJK4Fv 87A4TXXTJ[;>7A=?@HV MAB;UPTJ eK@H=?J[4S; B5V<JK9<hAJK9m7:6T6T9UM\7Ae PVUJKJ ^"DTVa;>M\4TJ[4CzA\&0 x bM J[hAJK9C4TM I7:;U;UJK9mPTMVUM\6TPF=?V<;U=?eK7:;>J[X ;UPTJrhi=?VUDT7A@?=1U 7:;U=?M\4V<MAB ; BN7A9<Js=?Vg;UPTJo6F9UMK5T@HJKI 9UJ[I7:=?4TVW;>P87 ; M\4TJ]e 7A4F4TMA; 9<J 7A@H@HnpV<JKJ =H4TVU=HXTJo7 ;>PT9<JKJ B5XT=HIJ[4TVU=HM\487A@MK5 <JKeu; x

&0

%2

!-2& * + "!-, /. 01 '

2mPT=?Vbe P87A6F;UJK9|eKM\4Te[JK4L;>9U7:;>J[V M\4 5TDT=?@HXT=?4FZ5 7A487A@ niVU=?Vb7A4FX`4SDTIJ[9U=He 7A@V<=?IsDT@?7:;>=HM\4 MABd71Z\JK4VBJK9>7:@8MAB`2ODT9U=H4TZQV<niV<;UJKIx 2mPT=?VlIMiXTJ[@T=?V e 7:@?@?J[XZAJK4TJ[9>7A@8XTDFJb;UM];>PFJb4T7:;>DT9<J|M:B ;>PFJ|9<J 7Aeu;>=?MA4ti=H4TJ[;U=?e[V WPT=He PI7AtAJm= ; 6CM\V<VU=15T@?JN;UMQVa;>DTXfn ;>PTJWJlXCJKeu; M:B XT=YX J[9UJ[4S;le[M\DT6T@H=?4TZ\V 7A4TX14TM\4T@H=?4.BJ 7A9 ;>J[9UI1VKx[2mPTJbI1MfXTJ[@ N7AVl=?4L;U9UMiXTDTeKJ[X=H4 =N7A9<9U=?MQJ[; 7A@ xBM0TKTKS! x\] J|DTV<Jw@H=?4TJK7A9 7A487A@ niVU=?V;>M16T9UJ[XT=?eu;|;UPTJZ5gJKPT7 hi=HM\9WMAB_;UPTJ]VaniV<;>J[I C7A4TX;>PTJQ4SDTI1JK9U=He 7A@I1J[;UPTMiXTVW6T9UJ[VUJK4L;UJKX =?4 ;>PFJ6T9UJuhf=HM\DTVe P87A6F;UJK9BEM\91VUMA@Hhi=?4TZ;>PFJ`V<niV<;UJKI DTVU=H4TZ e[M\I6FDF;>J[9V<=?IsDT@?7:;>=HM\4)x ?_=H487A@H@Hn ;>PFJ9UJKV<DT@H;UVb7A9UJr6T9<JKV<JK4L;>J[X 7A4TX7A487A@ nVUKJKX W= ;>Pp;>PTJYPTJ[@?6MAB V<;>9<DTe[;UDT9UJYB 7Aeu;>M\9<VKx

4&(' )4* , 63;*

?_=?9<V<; JYe[M\4TVU=HXTJK9N;UPTJrZ\J[4TJK9U7A@ BEM\9PTJ 5gJKZ\=H4T4T=H4TZ MAB[a P87A6F;UJK9bFxd3 ;b=?VmZ\= h\J[4 BEMA9N; Me PTJKI1=?eK7A@?V

U7:4TX

V7:V

Ut = DU∇2U + f(U, V )

Vt = DV ∇2V + g(U, V ),=J3TxY0!?

WPTJK9<JU ≡ U(~x, t)

7A4TXV ≡ V (~x, t)

7A9UJs;UPTJ VU687 ;>= 7:@?@Hn7A4FX;UJKI16CMA9>7A@H@Hn h:7A9<ni=?4FZpeKM\4.BeKJK4L;U9>7:;U=?M\4TV[x

2mPT=?VYV<niVa;>JKIP87:V]7 V<;UJ 7AXfnB V<;>7:;>J(Uc, Vc)

XTJ[GT4TJKX 5Lnf(Uc, Vc) = g(Uc, Vc) = 0

x 2mPTJJKRSD87:;U=?M\4TVYMAB I1MA;>=HM\47A9UJ MK5f; 7A=H4TJKX 5Ln6TPTJ[4TM\I1JK4TM\@HM\Z\=He 7A@ J@f687A4TV<=?M\4 MABl;>PTJ14TM\4F@?=?4FJ 7A99UJ 7:e[;>=HM\4 BEDT4Te[;U=?M\4FV

f7A4TX

g7A9<M\DT4TX1;>PTJbV<;>7:;>=HM\487A9ansDT4F=HBEM\9<IVUM\@HDF;>=HM\4

(Uc, Vc)xdqrJ[JK6T=H4TZ

Lz

;>JK9<IVmDF6;UM1e[DM5T=HeQMA9UXTJ[9 JQZAJ[;WBEM\9N;>PFJQ9<J 7Aeu;>=HM\4p;>JK9<IV

f(U, V ) = A(U − Uc) + B(V − Vc) − C(U − Uc)(V − Vc) − D(U − Uc)(V − Vc)2

g(U, V ) = E(U − Uc) + F (V − Vc) + C(U − Uc)(V − Vc) + D(U − Uc)(V − Vc)2,

WPTJK9<JA, B, C, D, E, F

7A9UJde[M\4TV<;>7A4L;>V;>Mf5gJdXTJu;>JK9<I=H4TJKX x\4bMA;>=HeKJ_;UP87:;;>PFJdeKMiJ1eK=HJK4L;>VMAB|;UPTJp4TM\4F@?=?4FJ 7A9o;UJK9<IV IsDFV<; 5gJpe PTM\VUJ[4 ;UM 5gJpM\6T6gM\V<=H;>Jp4SDTI 5CJ[9UV XTDTJ;UM;UPTJ eK9<M\VUVCB=?4TPT=15T=H;UM\9 B57Ae[;U=Hh:7:;UM\9N487:;UDT9UJ M:Bl;UPTJsVanfVa;>J[IxZ?_=Y@f=H4TZ

B = 17A4TX XFJK4TMA;U=?4TZ

A = α, C =

r2, D = αr1, E = γ, F = βni=?J[@?XTV

ut = Dδ∇2u + αu(1 − r1v2) + v(1 − r2u)

vt = δ∇2v + v(β + αr1uv) + u(γ + r2v),=J3Tx z0?

WPTJK9<Ju = U − Uc

7:4TXv = V − Vc

_I7AtS=?4TZp;UPTJ6gM\=H4S;(u, v) = (0, 0)

7 V<;>7:;>=HM\487A9anVUM\@HDF;>=HM\4)x\2mPFJrXT=1XCDTVU=HM\4e[M\4TV<;>7A4L;W=?VNXF=Hhi=?XTJ[XpVUMs;UP87:;

δ=?Vm7sV<e 7A@H=?4TZQB 7Ae[;UM\9m7A4TX

D=HV ;>PFJ

9>7:;U=?M5CJu; J[JK4p;>PTJ|XT=1XCDTV<=?M\41eKM\4FV<; 7:4S;UVNMAB);UPTJw; Mse PTJ[I=He 7A@HVKx 3 ; =?Vl=?I16CMA9<; 7:4S; ;UMs4TMA;U=?eKJ;>P87:;

D 6= 1=HVs9UJ[RSDT=?9<JKX BEM\9o;>PFJpXT=1XCDTVU=HM\4.B XT9U= h\JK4 =H4TV<;>75T=H@?=H;5n ;UM Mfe[eKDT9[x-] JpeKM\4TV<=?XTJ[9

u;UMH5gJs;>PFJ7:e[;>= h:7:;>M\9 7A4TX

v;>M 5CJs;UPTJ =?4TPF=65T= ;>M\9be PTJKI1=?eK7A@ )7A4TX;>PSDTVY9UJ[RiDF=?9UJ

D < 1x

2mPTJ 687A9U7AI1J[;>J[9UVr1

7A4TXr2

e 7:4e5CJ|DTVUJ[XBEM\9N7:X<DTV<;U=?4TZQ;>PTJ|4TM\4.B @?=H4TJ 7A9 =H4S;UJK9U7Ae[;U=?M\4TVF7:4TXα, β, γ, δ

7:4TXDBEM\9mI1MiXTJrVUJ[@?JKeu;>=HM\4 WPT=He P W=H@?@j5CJYXT=HVUeKDFVUVUJ[X"=H4 ;UPTJY4TJ@i;bVUJ[e[;U=?M\4)x

2mPTJeKMiJ1eK=HJK4L;>Vr1

7A4TXr2

VUJu;p;>PTJ 7AI16T@?= ;>DTXFJKVMABY;UPTJ4TM\4T@H=?4TJK7A9;>J[9UI1V WPT=?e P 7A9<JeK9UDFeK= 7:@ BEM\9 ;>PTJ I1M\9U6FPTM\@?MAZAn\x 3 ;1P87AV 5gJKJ[426gM\=?4L;>J[X M\DF; 5Ln =N7A9U9<=?M Ju;7:@ x 0TKTKS6N;>P87 ;=?4; M XT=?I1JK4TV<=?M\4FVNRSD87AXT9U7:;>=Hew=?4L;UJK9>7:e[;>=HM\4TV =

uv?deKMA9U9UJ[VU6gM\4TXp;>M V<6CM:;m687:;<;>JK9<4TV WPFJK9UJK7AV

eKDM5T=He=H4L;>JK9U7Ae[;U=?M\4FV =uv2 ?se[M\9U9<JKVU6gM\4TX ;>M Va;>9<=?6gJ687 ;U;>J[9U4TV[x = n ;>DT4T=H4TZ

r17A4TX

r2M\4FJ

e 7A4VUJ[@?JKeu;wJ[=H;UPTJK9WM\4TJQMAB ;>PTJY=?4L;UJK9>7:e[;>=HM\4TVWM\9WZ\J[4TJK9U7:;>J]71e[M\I16CJu;>=H;U=?M\4:5CJu; J[JK4`;>PFJY; M687:;U;UJK9<4TV#5Sn V<J[;<;>=?4FZ 5gMA;UP ;UJK9<IVm4TMA4MUKJ[9UMTx

4& % 1;* /+ /214/ . .

=H4TJ 7:9W7A487A@ nfV<=?VW=?VW7 I1J[;>PFMfXpBEM\9WV<;UDTXFni=?4FZ1;UPTJ]Va; 75F=?@?= ;5nM:Bd7 V<niV<;UJKI 7A9UMADT4TX7 Va;>JK7AXFn(BV<; 7 ;>JAx[= n @?=H4TJ 7A9 7:487A@HniV<=?V Jb;U9<n ;>MYG84TXIMiXTJ[VL= x JAxE N7 hAJ|h\J[e[;UM\9UVl;>P87:;lZ\9<M J@f6gM\4TJ[4.B;>= 7:@?@Hnp;UPSDTV]9<JKV<DT@H;U=?4TZp=H47A4=?4TVa; 75T=H@?= ;5n"7:4TXeKM\I16T@?J @ 687 ;U;>J[9U4 BEM\9<I7:;U=?M\4 x]354 7AXTXT= ;>=HM\4 Jr;>9an;UM1e PFMfM\V<Jr;>PTJr687:9>7AI1J[;UJK9UVm=H4pVUDTe P"7 N7 np;UP87:;mM\4F@Hne[JK9a; 7A=H4pXTJKV<=?9UJ[XI1MiXTJKV W=H@?@Z\9UMYx ?8M\9W7A4=?4L;>9<MfXFDTe[;U=?M\4;UM ;UPTJYIJu;>PTMiX VUJ[J 4|=HeKM\@H=?V0TKTK76M\9% ^"DT9<9>7 n0TT\! x

3 ;P87AV 5CJ[JK4 6gM\=?4L;UJKX M\Df;JK7A9U@H=?JK9 =N7A9U9<=?M J[; 7:@ xE 0TKTKS!b=?4 9UJ[@ 7:;U=?M\4 ;>M ;UPTJ"Va;>DTXF=?JKVMAB\

; M)BdXT=?I1JK4TV<=?M\4T7A@!2 DT9<=?4TZoV<niV<;UJKI1VT;>PT7:;N;>PFJK9UJwI7 nJ@f=?Va;W7s4SDTI 5gJK9NMAB XF=1XCJK9UJ[4L;mIMiXTJ[VP87 hf=H4TZY;>PTJbV>7AI1J N7 h\J|4SDTI 5gJK9Kx 2mPF=?Vl6TPTJK4FM\IJ[4TM\4=?V e 7A@H@?J[X -.012+' :x\2OMoI7AtAJW;>PFJeKM\4Te[JK6F;_eKM\4FeKJK= h:75T@?J e[M\4TVU=HXTJK9 ; Mw; MB5XT=HIJ[4TVU=HM\487A@\hAJKe[;UM\9UV ~k1 = (1, 0)

7A4TX ~k2 = (0, 1)x

y hAJK4;>PTM\DFZ\P ;>PTJ[VUJ h\J[e[;>MA9UV6CM\=H4L;;>M XT=YX J[9UJ[4S;XT=H9UJ[e[;>=HM\4TV;UPTJK=H9p@?J[4TZA;>P =HV;>PTJ VU7AI1J|~k1| = |~k2| = 1

x1354 ;UPT9UJ[J XT=?I1JK4TV<=?M\4FV ;>PTJ XTJ[Z\JK4TJ[9>7Aeun =?VrZ\9<J 7:;UJK9Q7A4TX Mfe[eKDT9<Vo7:@?9UJK7AXFn7:;wV<I7:@?@ N7 hAJs4SDTI 5gJK9UV[x a M\4FVU=?XFJK9|BEMA9wJ@F7AI16T@HJQhAJKe[;UM\9UV

(7, 1, 0), (5, 0, 5), (3, 4, 5)7A4TX

7A@?@ MA;UPTJK9|6CM\V<VU=15T@?JQ6CJ[9UIsDf; 7:;U=?M\4TVbP87 hi=?4TZ1;>PTJoV>7AI1J]@HJK4TZ:;>P5√

2x 2mPSDTVw;>PFJoRSDTJKVa;>=HM\4 MAB

WPT=?e P I1MfXFJ W=?@H@ XTM\I1=?487:;UJQ;UPTJsJuh\M\@HDF;>=HM\4 5gJKeKMAIJ[V IMA9UJoXF= 1eKDT@ ;|;UM 7A4TV J[9r=?4;UPT9UJ[JXT=?I1JK4TV<=?M\4FVKx

2mPTJ75CM hAJ 5gJKe[M\IJ[V MK5Lhf=HM\DTV WPTJ[42M\4FJ9<JKeK7A@?@HVo;>P87 ;=H4 7GT4T=H;UJpZ\9U=HX ;>PTJ7AXTI1=?V<VU=15T@?JIMiXTJKVW7:9UJY4TMA;We[M\4L;>=?4SDTMADTV#5TDF;bXT=?V<eK9<J[;>JQ7A4FX;UPTJYIMiXTDT@HDTVNMABO;>PTJ 7 h\JYh\JKeu;>M\9b=?VmZA=Hh\J[47AV

|~k| = k = 2π

√

(

nx

Lx

)2

+

(

ny

Ly

)2

+

(

nz

Lz

)2

,= 3Fx ?

WPTJK9<JLx

Ly

7A4TXLz

XTJK4FMA;>J ;UPTJ V<niVa;>JKI VU=6U[Jr=?49<JKV<6CJ[e[;>= h\JYXT=H9UJKeu;>=HM\4TVN7:4TXnx

ny

7:4TXnz;UPTJW9UJ[VU6gJKe[;U=HhAJ 7 h\Jw4SDTI 5CJ[9l=H4TXT=HeKJKV[x[2mPTJ[4;>PFJb@H=?4TJK7A9d7A4T7A@HniVU=HVd6T9UMieKJ[JKXTV 7AVdBEM\@H@?MWVKx

kwV 7AV I1JK4L;>=HM\4TJ[XoJ 7:9U@?=HJK9:y R x5=J3Tx z0?P87AV 7A4sV<;>7:;>=HM\487A9anYVUM\@HDF;>=HM\4oXFJ[G84TJ[X 5Ln];UPTJOUKJ[9UM\V_MABf7:4TX

gJ[h:7A@HD87:;UJKX7:;WV<;UJ 7AXfnB V<;>7:;>JYVUMA@?DF;U=?M\4

(u, v) = (0, 0)x = npV<M\@Hhi=?4FZs;>PTJYJKRSD87 ;>=?MA4TV

αu(1 − r1v2) + v(1 − r2u) = 0

v(β + αr1uv) + u(γ + r2v) = 0,=J3Tx 3?

JYMK5F;>7A=?47A4FMA;>PTJ[9mV<;UJ 7AXFn(B V<; 7 ;>JQVUM\@HDF;>=HM\4 7:;

v =−(α + γ)

1 + βu.

= 3Fx<7 ?

3541MA9UXTJ[9 ;>MrB 7AeK=H@?= ; 7:;UJl;UPTJb7A487A@ nfV<=?V JbJK4fBEM\9Ue[J(0, 0)

;>M 5gJm;>PTJmMA4T@HnoV<; 7 ;>=?MA487A9<n]DT4T= BEM\9UIVUM\@HDF;>=HM\4 5LnpVUJu;U;>=H4TZ

α = −γ8;>PSDTVWI7:ti=H4TZs;>PFJQ9<JK@?7:;>=HM\4pMAB ylR x8=J3Tx<70?lMK5TVUMA@?J[;UJAx

v =?V<6CJ[9UVU=HM\4p9UJ[@ 7:;U=?M\4=?Vm7:4=HI6gM\9a; 7A4L; eKM\4Te[JK6F;b9<JK@ 7 ;>=?4FZ];>M @H=?4TJK7A9N7:487A@HniV<=?VKx_3 ;mnf=HJK@HXTV ;>PFJJK=?ZAJK4Lh:7A@?DTJ[V MABC;UPTJb@?=?4FJ 7A9<=6UKJ[X JKRSD87:;U=?M\4FV 7:V 7rBEDT4Teu;>=HM\4 MAB 7 h\Jw4SDTI 5CJ[9Kx 354 ;>PTJ|75TV<JK4Te[JMAB_XF=1XCDTVU=HM\4;>PTJY@H=?4TJK7A9U=1UKJKXpVaniV<;>J[I =?VmZ\= h\JK47AV

(

ut

vt

)

=

(

fu fv

gu gv

)

uc,vc

(

u

v

)

.

WPTJK9<Jl;UPTJ V<DM5.B =?4TXTJ[eKJ[V_V<;>7A4TXQBEM\9 687A9<;U= 7A@AXTJK9<=Hh:7:;>=HM\4Q7A4TXY;>PTJ[VUJ XTJK9<=Hh:7:;>= h\J[V 7:9UJlJ[h:7A@?DT7:;>J[X

3

7:;o;UPTJV<;UJ 7AXfn V<;>7:;>J(uc, vc) = (0, 0)

x`354 ;UPTJ6T9<JKVUJ[4TeKJ MABWXF=1XCDTVU=HM\4 W=H;UP ;>PFJ687A9a;>= 7:@XTJK9<=Hh:7:;>= h\J[VmMAB ;UPTJY9UJK7Ae[;U=?M\4 tS=?4TJu;>=HeKVWe 7A@HeKDT@?7:;>J[X JQMK5F;>7A=?4

(

ut

vt

)

=

(

δD 0

0 δ

)(

∆u

∆v

)

+

(

α 1

γ β

)(

u

v

)

.

4|M%A@HJ[;uk(~r, t)

7A4TXvk(~r, t)

5gJ ;>PTJlJK=HZ\JK4FBEDF4Te[;U=?M\4TV e[M\9U9<JKVU6gM\4TXF=?4TZb;>MW;>PTJ N7 hAJN4SDTI 5gJK9k7A4TX J 7Ae P BEDT4Teu;>=HM\4 VU7:;>=HVPTJ 5CM\DF4TX87A9an e[M\4TXT= ;>=?MA4TVKx ] JVUJKJ[t BEM\9QVUMA@?DF;U=?M\4TV]MABBEM\9UI

u(~r, t) =∑

k

ckeλtuk(~r, t)

=J3Tx98 ?

v(~r, t) =∑

k

ckeλtvk(~r, t),

=J3Txb; ?

WPTJK9<Js;>PTJse[M\4TVa; 7A4L;>Vck7A9<JsXTJu;>JK9<I=H4TJKX 5Ln;>PTJs=H4T=H;U= 7A@e[M\4TXT= ;>=HM\4TVKx%] PFJK4 ;>PTJ[VUJ V<M\@?D.B

;>=?MA4TVW7A9UJrV<DM5TVa;>=H;UDF;>J[X;UM1y R x8= 3Fx z ? J]MK5f; 7A=H4(

λuk

λvk

)

=

(

δD 0

0 δ

)(

−k2uk

−k2vk

)

+

(

α 1

γ β

)(

uk

vk

)

.

2mPTJYJK=?ZAJK4Lh:7A@?DTJ[Vλ7A9UJYXTJu;>J[9UI1=?4TJ[XH5Ln;UPTJYe P87A9U7Ae[;UJK9U=HV<;U=?eY6CMA@Hni4TM\I1= 7A@

|λI − A + Dk2| = 0,=J3TxbS ?

WPTJK9<JIXTJ[4TMA;UJKV ;>PTJb=?XTJ[4L;>=H;5nsI7:;U9U= @;

A;UPTJ|@H=?4TJK7A9U=1UKJ[X I7:;U9U= @ 7A4TX

D;UPTJbI7:;U9U= @ W=H;UP

;>PTJYXT=YX DTV<=?M\4pe[M\4TVa; 7A4L;>V[xOaN7:@?eKDF@ 7:;U=?M\4M:B ;UPTJYXTJ[;UJK9UI1=?4T7A4L;mnf=HJK@HXTV

λ2 − (α + β − δk2(1 + D))λ + (α − δDk2)(β − δk2) + α = 0,=J3Tx9T ?

WPTJK9<Jk2 = ~k · ~k x

2mPTJ eKM\4TXF=H;>=HM\4TVwBEMA9 XT=1XCDTV<=?M\4.B XT9U= h\J[4 =?4TVa; 75T=H@?= ;5np;>M Mfe[eKDT9Y7:9UJ W=?XFJK@Hn"tS4TMW4 ^"DT9<9>7 n@0TKT\! ;>M 5CJ ;UPTJrBEM\@?@HMW=?4TZ

fu + gv < 0= 3Fx10 ?

fugv − fvgu > 0= 3Fx10K06?

Dugu + Dvfu > 0= 3Fx10 z0?

Dugu + Dvfu > 2√

DuDv(fugv − fvgu),= 3TxY0 ?

(7

WPTJK9<Jw;>PTJ|687A9a;>= 7:@8XTJK9<=Hh:7:;>= h\J[VN7A9<J|eK7A@?e[DT@ 7:;UJKX7:;(uc, vc)

7A4TXDu = δD

7:4TXDv = δ

7A9<J;>PTJYXT=YX DTV<=?M\4pe[MfJ1eK=?J[4L;>VWBEM\9me PTJKI1=?eK7A@?V

u7A4TX

vT9<JKVU6gJKeu;>= h\JK@ n\x

? 9<M\I ;>PTJs75CM hAJ eKM\4TXF=H;>=HM\4TV JseK7A4 eKMA4TeK@HDTXTJo;UP87:; ;UPTJsM\4TV<J[;YMAB =H4TV<;>75T=H@?=H;5n=?VwV<DM5 <J[e[;;>M ;>PTJrBEM\@H@?MW=?4FZseKM\4FXT=H;U=?M\4TV[j =HB

α ≥ 0T;UPTJK4

−1 < β < −α=J3TxY03?

α − 2√

αD > βD,= 3Fx1070?

7A4TX =HBα < 0

f;>PTJ[4

β < min−1,−α =J3TxY08 ?α − 2

√αD > βD (α > βD)

=J3Tx10; ?α + 2

√αD > βD (α < βD).

=J3Tx10S ?

= n"9<JKVa;>9U=He[;U=?4TZ;UPTJs687:9>7AI1J[;UJK9wV<JK@HJKe[;U=?M\4"VUDTe P ;>PT7:;α ∈ (0, 1)

7A4TXβ ∈ (−1, 0)

J 7A9UJ@?J[B ; W= ;>P M\4T@ n; Me[M\4TXT= ;>=HM\4TV

β < −α= 3Fx10T ?

α − 2√

αD > βD.= 3Fx zA ?

2mPTJ XT=?V<6CJ[9UVU=HM\4 9UJ[@ 7:;U=?M\4λ(k)

e 7:4 5gJp6T@HMA;U;UJKX 5Ln DTV<=?4TZ y R x= 3Tx9T ?>x-2mPTJ MK5 <J[e[;U=Hh\J =HV;>MpG84TX 687A9>7:IJu;>JK9 h:7A@HDTJKVr;UP87:;Ye 7:4I5gJ DTVUJ[X ;>M=?V<M\@ 7 ;>JsI1MfXFJKVKx ?_=HZ\DT9<J 3Tx10 V<PTMWV];>PFJ9UJ 7:@C687:9<;>V MAB;>PTJwJK=?ZAJK4Lh:7A@?DTJ[VN6T@HMA;U;UJKXp7AZ\7A=?4TVa; ;UPTJ 7 h\JY4SDTI 5gJK9

kBEM\9 ;UPT9UJ[J IMiXTJ[VKxd354

7AXTXT= ;>=?MA4 J DTVUJ[X 7:4 J[RSD87:;>=HM\4`BEM\9w;UPTJ eK9<=H;U=?e 7:@ 7 h\J14iDFI 5CJ[9= x J:xB ;UPTJ 7 hAJ4SDTI 5CJ[97:;m;>PTJYI7@f=?IsDFI MABO;>PFJQe[DT9<hAJAx 2mPT=?VW=HVmZ\=HhAJK4H5Sn

k2c =

D(α − β) − (D + 1)√

αD

δD(D − 1).

= 3Tx z.0!?

3 ;N=?V M\9<;UP4TMA;U=?eK=H4TZ];UP87:;l;>PTJ 7A@?Z\Jl5T9>7:=?emBEM\9UIM:B ylR x2= 3Tx9T ? I7At:JKV =H; XT= 1eKDF@H;l;>Ms7:487A@HnVU[J;>PTJYV<niVa;>JKIBEDF9<;>PFJK9Kx 2mPT=HVW=?V WPLn y Rgx8=J3Tx z.0!?l=?VNhAJK9an DTVUJuBEDT@ x

4& , / : (671 + * . (:<.

2mPT=?V V<JKeu;>=?MA4`6T9UJ[VUJ[4S;UVr;>PTJs9<JKV<DT@H;UV MABd;>PFJo;>PT9<JKJ B5XT=HIJ[4TVU=HM\487A@ VU=HIsDT@?7:;>=HM\4TVKxbv J[VU6T= ;>Js;>PFJB 7Ae[; ;UP87:; ;>PTJ Va;>9UDFe[;>DF9UJKV7A6T6gJ 7:91=HI6F9UJKV<VU= h\J< JeK7A4T4TMA;1XT9U7 6T@?7ADTVU=15T@?JeKM\4Te[@?DTV<=?M\4FV

K8

0 0.5 1 1.5 2

−0.2

0

λ (k

)

wave number k

?_=?Z\DF9UJ 3Tx10Aj[2mPTJrXT=HVU6gJK9UV<=?M\49UJK@?7:;>=HM\41MAB y Rgx= 3Tx z ? BEMA9 VUJ[@?J[e[;>J[XpIMiXTJ[VKx 2mPTJ 6T7A9>7AI1J[;UJK9<V J[9UJ

D = 0.516α = 0.899

β = −0.91

7A4TXδ = 2

BEM\9k = 0.45

= X87AVUPVB5XTMA;o@?=H4TJ6?D = 0.122

α = 0.398

β = −0.4

7A4TXδ = 2

BEM\9k = 0.84

= XTM:;U;>J[X @?=H4TJ6?D = 0.516

α = 0.89

β = −0.99

7A4FXδ = .25

BEM\9k = 0.96

= V<M\@?=HX @H=?4TJ ? x 2mPTJ`9<JKZ\=HM\4 75CM hAJλ(k) = 0

5gM\DT4TXFVW;UPTJ N7 hAJo4SDTI 5gJK9mh:7A@?DFJKVNBEM\9mDT4FV<; 7)5T@?JYI1MfXTJ[VKx

(;

587AVUJ[X MA4 7pRSD87A@H=H;>7:;>= h\J 7A4T7A@HniVU=HV7:@H;>PFM\DTZ\P`;UPT=?Vr=HVr6T9UMK5T75T@Hn;>PTJ 5gJKVa; N7 n`;UM7:487A@HnVU[J;>PTJ9UJ[VUDT@ ;>V[x k|4Sn 7 n J ;U9<n RSD87A4L;U=H; 7 ;>=HhAJ 7A487A@ nfV<=?V 5Ln XTJu;>JK9<I=H4T=?4FZ ;>PTJV<;U9UDTeu;>DT9<JB 7Ae[;UM\9UVmMABO;>PFJQVa;>9<DTe[;UDT9UJ[VKx

354";>PTJ]BEM\@?@HM W=H4TZ5 WPTJ[4 JoXT=?V<eKDTV<Vr75gM\DF;w;UPTJs7:XTXT=H;U=HhAJo4TM\=HVUJ! JsI1J 7A4 _Y7ADTV<VU=?7A4"DT4VBeKM\9<9UJK@?7:;>J[X 4TM\=?V<J WPT=He PP87:V 5CJ[JK47AXTXTJ[X1;UMQJuh\J[9<n VU= ;>JW7:; J[h\J[9<n ;U=?I1JNV<;UJK6)x 4 ;>PTJmM:;>PTJ[9P87A4TX5Sn 78DTeu;>D87:;U=?M\4FV J I1J 7:4`6CJ[9<;UDT9 587 ;>=?MA4TVw;>PT7:;r7A9<J 7AXTXFJKX ;>M;UPTJsG8487:@_V<;U9UDTeu;>DT9<J;>MMK5TV<JK9<hAJ"= ;>V V<;>75T=?@H=H;5nAx 2mPTJ G89UVa;7A4FX V<JKeKMA4TX2I1M\I1JK4L;>V MABb;>PTJ 4TMA=?VUJ7A9<J Z\=HhAJK4 5Sn〈η(~x, t)〉 = 0

7A4FX 〈η(~x, t)η(~x′, t′)〉 = 2εδ(~x − ~x′)δ(t − t′)x 2mPTJ 7A4FZ\DT@ 7:9:5T9U7Ae tAJu;>V

XTJK4TM:;>JQ7A47 hAJK9U7AZ\JQ7A4TXε=HVN;>PTJY=?4L;UJK4TV<=H;5npMABO;>PFJQ4FM\=?V<JAx

> >B? DAR RD F T PQJ J PWGIT

?_=?9<V<; Jme[M\I1687A9UJ ;>PTJO5gJKPT7 hi=HM\9 MAB8;UPTJmV<niV<;UJKI=H4o;>PF9UJKJmXF=?I1JK4TV<=?M\4TVO;>M 9<JKV<DT@H;UVdMK5F; 7:=?4TJ[XBE9UM\I7YVU=HI6F@?JN; M)B5XT=HIJ[4TVU=HM\487A@FVU=?IoDT@ 7:;U=?M\4 =N7A9<9U=HMQJu;N7:@ xEM0TKTKS6 x\?_=?Z\DF9UJ#3Tx zYV<PTMWV ;>PFJ9UJK6F9UMiXTDTeKJ[X 9UJ[VUDT@ ;>V =H4; MpXT=?I1JK4TV<=?M\4FVKxb354`;>PTJQ;>M\6`9<M%C;>PFJs687A9U7AIJu;>J[9UV JK9<J e PTM\V<JK4;>MpB 7 h\M\9QV<;U9U=?6gJKX687 ;U;>J[9U4TV = x J:xB

r2 = 07:4TX =H4 ;>PTJ 5gMA;U;UM\I-9UM&VU6gMA;U;5n 6T7:;U;UJK9U4FVO= x JAxE

;>PTJ ;UJK9UIr2

N7:VmVUJ[;bXTMAI=H487A4L; x\?_=?Z\DF9UJ%3Tx sVUPFM WV W= ;>P;>PFJrV>7AI1Jr@ 7 n\MADF;W7AV#?_=HZ\DT9<J%3Tx z;>PTJme[M\9U9<JKVU6gM\4TXF=?4TZYV<;U9UDTeu;>DT9<JKV MK5F;>7A=?4FJKXsBE9UMAI;>PFJmVU=?IoDT@ 7:;U=?M\4FV MAB ;UPTJN;>PT9<JKJ B5XT=HIJ[4TVU=HM\487A@V<niV<;UJKIx

kwVeK7A4 5CJ"4TMA;>=HeKJ[X BE9UM\I ;>PFJ";>M\6 9<M%M:B?_=?Z\DF9UJ^3Fx z ;UPTJK9<J7A9UJ"M\4T@Hn 7 BEJ XFJ[BEJKeu;>V= x JAxEg6gM\=H4S;UV WPFJK9UJQ;>PTJQV<;U9U=H6CJ[V|e[M\=?4Te[=?XTJ:x |M J[h\J[9C=?4 ?_=?Z\DF9UJKV 3Tx \717A4TX^5"VUPTMW=?4FZ;>PFJ@ 7AI1JK@H@ 7A9 V<;>9<DTe[;UDT9UJ[VF7]Z\9<J 7:;UJK9 4SDTI 5gJK9 M:B)XTJuBEJKe[;UV ;UP87A4=H41;UPTJbV<;>9<=?6gJKX1; M)B XT=?I1JK4FVU=?MA487A@687:;U;UJK9<4TV]eK7A4 5CJ VUJ[JK4)x:2mPT=?VY=?VYXTDFJ1;UM ;UPTJ1XT= 1eKDF@H;5n MAB 7A@H=?Z\4T=H4TZ6T@?7A4TJ[V]e[M\I6T7A9UJ[X;UMV<;>9<=?6gJKVm=H4687A9U7A@?@HJK@ x2mPTJrXT9<M\6T@?Ju;W687:;<;>JK9<4TVm=?4;UPTJ 5gMA;U;UM\I 9<MMAB?_=HZ\DT9<J%3Tx zs7A4TX V<;U9UDTe B;>DT9<JKVO=?4?_=?ZADT9UJ[VQ3Fx Ael7A4FX]X]7A9UJ I1M\9UJ V<=?I1=?@ 7:9CXTJ[VU6T= ;>J ;>PTJl=?4FeK9UJK7AVUJ =H4Y;UPTJlXT=?I1JK4TV<=?M\4T7A@?= ;5nMABN;>PFJV<niVa;>JKIx bM Juh\JK9 =H;Qe 7A4-5CJVUJKJ[4 ;>P87 ; WPTJ[9UJ 7:Vo;>PTJ1; MB5XT=HIJ[4TVU=HM\487A@dVanfVa;>J[IV<; 7)5T=?@H=6UKJ[V|;UMp7PTJ@F7AZAM\487A@ @ 7:;<;>=?e[J<g;>PT=HVwtS=?4TX M:B 6FDT9UJsVa;>9<DTe[;UDT9UJ e 7:4T4TMA; 5gJsMK5FVUJK9ah\J[X =H4;>PT9<JKJ1XT=?I1JK4TV<=?M\4FVQXFDTJ;UM;UPTJ=H4TeK9<J 7AV<JKX 4SDTI 5CJ[9oMAB XTJKZ\9<JKJ[VoMABNBE9<JKJ[XTM\Ix354 ;UPTJ4FJ@i;VUJKeu;>=HM\4 J W=?@H@ XT=HVUeKDFVUVbPTM;>M MK5f; 7A=H49<JKZ\DF@ 7A9NVa;>9<DTe[;UDT9UJ[VKx

4|MA;U=?eKJ ;>P87 ;";>PFJ ; M IMiXTJKVk = 0.45

7A4TXk = 0.84

XT=1XCJK9RSD87A@?= ; 7:;U=HhAJK@Hn+5Sn;>PFJN7 h\J[@?J[4TZA;>P

λ = 2π/kx"kwV

k=?4FeK9UJK7AVUJ[V ;UPTJ 7 hAJK@HJK4TZA;UP XTJKe[9UJ 7:VUJKV 9<JKV<DT@H;U=?4TZ=?4 687 ; B

;>JK9<4TV W= ;>P @?JKV<VoXT=HV<;>7A4TeKJ 5CJu; J[JK4 Va;>9U=H6CJ[Vs7A4FX @ 7:9UZ\J[9]4SDTI 5gJK9QMABN;UPTJVa;>9<=?6gJKVKx354 ; MXT=?I1JK4TV<=?M\4FVW;UPTJoVU7AI1JoJlXCJKeu; e[M\DT@?X 5CJo7A@?V<MMK5f; 7A=H4TJKXH5Ln"e P87:4TZ\=?4FZ;>PFJoVUeK7A@?=H4TZ B 7Ae[;UM\9

δ=?4 5gMA;>P JKRSD87:;U=?M\4TV[x |M J[h\J[9l=?4 ;UPT9UJ[JXT=HIJ[4TVU=HM\4TVY;>PTJRSD87:@?=H;>7:;>= h\J1XT=1XCJK9<JK4Te[JpMABm;>PFJ

KS

?_=?Z\DF9UJ 3Tx zfj _7:;U;UJK9<4TVwMK5f; 7A=H4TJKX BE9UM\I VU=?IoDT@ 7:;U=?M\4FVNMAB_;UPTJQZ\JK4TJ[9>7A@2ODT9U=H4TZI1MiXTJK@=H4 ; MXT=?I1JK4TV<=?M\4FVbDFVU=?4FZ9U7A4TXTM\I =H4T=H;U= 7A@CeKM\4FXT=H;U=?M\4TV[xE2OM\6"9<MYj

r1 = 3.57A4TX

r2 = 0 5gMA;<;>M\I9UMYjr1 = 0.02

7A4TXr2 = 0.2

x 354 ;UPTJm@?JuB ;deKMA@?DTI14:;UPTJm687A9U7AI1J[;>J[9UV JK9<Jbe PFM\VUJ[41;UM B 7 hAM\9k = 0.45 =H4 ;>PTJ 9U=HZ\PL;YeKM\@HDTI14;UMpB 7 hAM\9YIMiXTJ k = 0.84

x AWJK6T9<MfXTDFeKJKXBE9UMAI =N7A9<9U=?MJ[;|7A@ xE0TKTKS! x

KT

?_=?Z\DF9UJ 3Tx Fj> 7 ;U;>J[9U4TV]MK5F;>7A=?4TJ[X BE9<M\I VU=HIsDT@ 7 ;>=?MA4TVrMABN;UPTJZ\J[4TJK9U7A@ 2 DT9<=?4TZIMiXTJ[@ 7:B ;>J[9500 000

=H;UJK9U7:;>=HM\4TVr7A4FX DFVU=?4FZp9>7A4TXFM\I =H4T=H;U= 7A@ eKMA4TXT=H;U=?M\4FV M\4J[hAJK9<nV<=H;UJ MAB ;>PTJ @ 7 ;U;>=HeKJ:ju0, v0 ∈ (0, 1)

xf2 MA6"9UMYjr1 = 3.5

7:4TXr2 = 0 5gMA;<;>M\I9<MYj r1 = 0.02

7A4FXr2 = 0.2

x JuB ;e[M\@?DTI14 ;UPTJ"6T7A9>7AI1J[;UJK9<V JK9UJ e PFM\VUJ[4 ;>M B 7 h\MA9

k = 0.45 =?429<=?Z\PL;eKM\@HDTI4 ;>MB 7 hAM\9sI1MfXFJk = 0.84

x-a MAI687:9UJ;UPTJp9UJ[VUDT@ ;>V W= ;>P ;>PFJ; M)B XT=?I1JK4FVU=?MA487A@l687:;U;UJK9<4TV =?4?_=?Z\DF9UJ%3Tx zfx

; MpRSD87A4L;U=H; 7 ;>=HhAJK@ n XT=1XCJK9<JK4L; eK7AVUJ[Vr=?VbeK@?JK7A9UJ[9>= ?_=?Z\DF9UJKV 3Tx \77A4FX 52?/CVU=H4TeKJ WPT=?@HJs7@H=?4TJP87AVWM\4T@ n;>PFJQV<@?M\6gJrXTJu;>JK9<I=H4T=?4FZs;>PTJYXT=H9UJKeu;>=HM\4T7 6T@?7A4TJrP87:VW; Mh:7A9<= 75F@?JKVN7A@?=HZ\4T=?4FZo=H;Kx

2mPTJe PT7A4TZ\J1=?4 ;>PFJ 7 h\J[@?JK4FZA;>P PT7AVr;>PTJ V>7:IJ JlXCJKeu;o=H4 ;>PTJ1e 7AV<J MABNXT9<M\6T@HJ[;Y687:;<;>J[9U4TV[x2mPTJXF9UM\6T@HJ[;UV 5CJ[eKM\I1J VUI7A@?@HJK9r7A4FX;UPTJ1XT=?Va; 7A4Te[J 5CJu; J[JK4 ;UPTJKI V<PTM\9a;>JK9[x 3 ;]PT7AV 5CJ[JK46CM\=H4L;>JKX MADF; 5Ln =N7A9U9<=?M Ju;7A@ x)Y0TKTKS!|;>P87:; =H4 ; M XT=HIJ[4TVU=HM\4TV;UPTJ VU6gMA;UV 7A9UJ I1M\9UJ9UMK5TDFV<;W;>P87:4Va;>9U=H6CJ[VKx 2mPTJQ9UMK5TDFV<;>4FJKVUV|IJK7A4TVm;>PFJQVa; 75F=?@?= ;5nM:B_;>PFJQ6T7:;U;UJK9U4M\9WV<;U9UDTeu;>DT9<JW=H;>P 9<JKVU6gJKeu;;UM6CJ[9<;>DF9 587:;U=?M\4FVlXT=1XCJK9<JK4L;1=?4T= ;>= 7:@ e[M\4TXT= ;>=?MA4TV M\9 MA;UPTJK9 e P87A4FZ\JKV1=?4 ;>PFJVU=?IoDT@ 7:;U=?M\4 e[M\4TXT= ;>=HM\4TVKx 2mPFJ9UM5TDTV<;U4TJKV<Vse 7:4 5CJ1V<;UDTXT=HJKX J:x ZTx 5Ln 7AXTXT=H4TZ4TM\=?V<JAx 2mPT=HVW=?@?@ 5gJ`XT=?V<eKDTV<VUJ[X @?7:;>J[9KxkwVpMA4TJ M\DT@?X J @F6gJKeu; b;>PTJ`V<6CM:;>V;>DF9U4TJ[XMADF;;>M-5CJ I1M\9UJ9UMK5TDFV<;]7A@?V<M =?4;>PT9<JKJ1XT=?I1JK4TV<=?M\4FVKx 2mPT=?VYeK7A4I5gJ1J@f6CJ[e[;UJKX-5T7AVUJ[X M\4;>PTJ B 7Aeu;Q;UP87:;Y;>PFJ

3L

;>PT=H9UX"XF=?I1JK4TV<=?M\4XTMfJ[V 4TMA; 5T9U=H4TZp7AV|IsDTe P 4TJ ;>M;UPTJsVU6gMA;<;5n"Va;>9UDFe[;>DF9UJKVY7:Vw=H;wXTMiJKV|;UM;>PTJY@ 7:IJ[@?@ 7:9 Va;>9UDFe[;>DF9UJKV[x

2mPTJrG8Z\DT9<JKVW=?I16T@Hn ;UP87:;WV<;>7A9<;U=?4TZsBE9<M\I e[M\I16T@?Ju;>JK@ n9U7A4TXTMAI =?4F=H;>=?7A@ eKM\4FXT=H;U=?M\4TVT=H;N=?VN4TM:;@?=?t:JK@ n]BEM\9 ;UPTJbV<niV<;UJKI;>MYe[M\4Lh\JK9<Z\Jw=H4L;>MQ7r6TDT9UJ[@Hn @?7AI1JK@?@?7A9OV<; 7 ;>JAx\2mPT=?V 7AV eKM\4FGT9UI1JKX 5Sn714SDTI 5gJK9|MAB J @i;>JK4FVU=HhAJsVU=HIsDT@?7:;>=HM\4p9UDT4FVKxf2OMI7AtAJQ=H;We[@?J 7:9 4TMA;U=?eKJY;>PT7:;W;>PTJ]XT=1XCJK9<JK4Te[J5CJu; J[JK4 ? =?Z\DT9<J\3Tx \7m7A4FX5Y=HV ;>P87 ;;>PTJd687:9>7AI1J[;UJK9UV

α7A4TX

βPT7 hAJ 5CJ[JK4QVUJK@HJKeu;>JKXY=H4rVUDTe P]7

N7 no7AV_;>MwJK4TP87:4TeKJNI1MiXTJKVk = 0.45

7:4TXk = 0.84

xQ2mPTJ[VUJW7A9<J ;>PTJ IMAV<; DF4TV<;>75T@?J IMiXTJ[V=?4 ;>PFJdVUJK4FVUJlMABf;UPTJd@?=H4TJ 7:9)7A6T6F9UM@f=?I7:;>=HM\4wBEM\9)6T7A9>7AI1J[;UJK9V<J[;>V

α = 0.899β = −0.91

δ =

2D = 0.516

f7A4TXα = 0.398

β = −0.40

δ = 2

F7A4TXD = 0.122

F9<JKV<6CJ[e[;>= h\J[@Hn\x 2mPFJKVUJ687A9<;U=?e[DT@ 7A9e PFM\=?e[JKV_M:B8687A9U7AIJu;>J[9UV J[9UJNI7AXTJl=?4QM\9UXFJK9 ;UMbB 7:eK=?@H=H;>7:;>J ;UPTJ e[M\I6T7A9U=HVUM\4 W=H;UP;>PTJ]z:v 9UJ[VUDT@ ;>VKx 2mPTJQ9UJK7AVUMA4BEM\9N;UPTJKV<J]e PFM\=?e[JKVb=?VN;>PT7:;b=?4p;>PTJrG89<V<;be 7:VUJ< =H4 ;UPTJYVUJ[4TVUJ]MAB;>PTJQ@?=H4TJ 7A9b7A4T7A@HniVU=HVT;UPTJK9<Js7A9<JQhAJK9<n BEJ 7AXTI1=?VUV<=65T@HJrIMiXTJ[V W= ;>P6CM\V<=H;U=Hh\JYZ\9<Mm;>P`9U7:;>J[V7A4TX:U[JK9<M 7 hAJK4SDTI 5gJK9 XTJKZ\J[4TJK9U7Ae[nAx kwVl=H;l=?V eK@HJ 7A9 BE9UMAIy Rgx2= 3Fx ?SXTJKZ\J[4TJK9U7Ae[n 5gJKeKMAIJ[V=?4Te[9UJ 7:VU=?4FZ\@Hn=?I16CMA9<; 7:4S;N7:;WPT=HZ\PTJK9 7 h\JQ4SDTI 5gJK9UV[x 35XTJ 7A@H@Hn<f;>PFJQe PFM\=?e[J

k = 0.45VUPFM\DT@?X

6T9UMiXTDTe[J687 ;U;>J[9U4TV W=H;UP 7 h\J1h\JKeu;>M\9 ~k = 2π(nx/L, 0/L, 0/L) = x JAxE B 7 hAM\9U=H4TZV<;>9<M\4TZ\@ n

;>PTJ @ 7:IJ[@?@ 7:9w6TP87:VUJ WPTJK4r2 = 0

x bM J[hAJK9d7AVYVUJ[JK4 =H4-? =?Z\DT9<J 3Tx \7 );>PFJVanfVa;>J[I PT7AV4TMA;m9<J 7Ae PTJ[X7o@ 7AI1JK@H@ 7A9 Va; 7:;UJ J[hAJK47:B ;>J[9

500 000;>=HIJ|V<;>J[6TVf5TDF;N=H;NPT7AVNeKMA4ShAJK9<Z\JKX ;>M 7

I= @FJ[X V<; 7 ;>JY=?4TVa;>JK7AX)x 2mPTJQVU=HIsDT@ 7 ;>=?MA4TV e[M\4FG89<IJ[X ;UP87:;W;>M 5gJr;>PTJYV<;>75T@HJ G8487A@Va; 7:;UJAx

?_=?Z\DF9UJ 3Fx 5pXF=?VU6F@ 7 niVN;>PTJrV<=H;>DT7:;>=HM\4 WPTJ[9UJr;>PFJrIMiXTJk = 0.84

=?VlB 7 hAM\9UJ[X)x 354p;>PT=HVNe 7AV<J<;>PTJ[9UJ7A9UJ1IMA9UJ eK@HM\VUJ[@Hn V<687AeKJ[X 7AXTI1=?V<VU=15T@?J I1MfXFJKVY@?JK7AXT=?4FZ ;UM"7eKM\I16gJ[;>= ;>=HM\4 5CJu; J[JK4;>PTJ[IxO3 ; =HV =HI6gM\9<;>7A4L;;>M|4TMA;>=HeKJ ;>P87 ;_;>PFJ VanfVa;>J[I&P87AV_7bG84F=H;>JlVU=1UKJ 7A4TX];UPSDTV_;>PFJK9UJ 7A9UJ!A=H4;>PTJbVUJ[4TVUJ|MAB);>PFJ|@H=?4TJK7A9 7A487A@ niVU=?VLM\4T@ n eKJK9a; 7A=H47:XTI=HVUV<=65T@HJNIMiXTJ[V)= ylR x@= 3Fx ?'? x ?89<M\I&;>PFJ;>M\6TI1M\Va;mG8Z\DT9<JKVbMAB[? =?Z\DT9<J 3Tx = ;W=?VWeK@HJ 7A9N;UP87:;m;UPTJ]e[M\I16CJu;>=H;U=?M\4:5CJu; J[JK4";UPTJYIMiXTJKVW=H4;>PTJm;>PF9UJKJ B5XT=HIJ[4TVU=HM\[email protected] DF9U=?4FZQVaniV<;>J[I e 7A4 @?JK7AX ;>MYh\J[9<n =?4L;UJK9UJ[V<;U=?4TZYI1M\9U6TPFM\@?M\ZA=?JKV[x2mPTJXTJ[BEJ[e[;>V =H4o; MYXT=HIJ[4TVU=HM\4TV eK7A4 5gJmeKM\4TV<=?XTJ[9UJ[X7:V 9UJ[I=H4T=?V<eKJK4L;UVdMAB8;>PTJmI1M\9<JNeKM\I16T@H=?e 7 ;>JKX687:;U;UJK9<4"VUJ[@?J[e[;>=HM\4"=H4":v&VU=H4TeKJQ;>PTJoz:vVa;>9<=?6gJQ6T7:;U;UJK9U4FVwe 7:4^5gJoVUJ[JK4 7:V|e[DF;>VbBE9UM\I 7AvV<niV<;UJKIx bM J[hAJK9_7AVwVUJKJ[4 =H4 ?_=HZ\DT9UJ[V%3Tx \77A4FX 5; ;>PTJ :ve 7AV<J XT=?V<6T@ 7 niV IsDTe P9<=?e PTJ[95CJ[P87 hf=HM\9s7:4TX =H;o=?VQh\JK9an XF= 1eKDT@ ;o;UM`MK5F;>7A=?4 7"6TDT9<JK@Hn @ 7:IJ[@?@ 7:9 687:;<;>JK9<4 V<;>7A9<;U=?4TZBE9UM\I9>7A4TXFM\I =H4T=H;U= 7A@geKM\4FXT=H;U=?M\4TV[x

354o;>PTJ 5CM:;U;>MAI9UM MAB?_=HZ\DT9UJ 3Tx ;>PTJN6T7A9>7AI1J[;UJK9<Vr1

7A4TXr2

P87 h\JO5gJKJ[4 VUJK@HJKeu;>JKX];>MwB 7 h\MA9VU6gMA;>ViMA9lV<6TPTJ[9U=?eK7A@ XT9UMA6T@?Ju; VUP87A6gJKV = ;UPTJKV<JweK7A4 5gJWeKM\I1687A9<JKX1XT=?9<JKeu;>@Hno;>M?_=?ZADT9UJ[V z J|7:4TXz Bl=?4 =N7A9U9<=?MJu;Q7A@ xEQ0TKTKS!C? x2mPTJ XT9<M\6T@HJ[;r687 ;U;>J[9U4TV ;UDT9U4FJKX M\DF; ;UM 5CJshAJK9<n`9UMK5TDFV<; x]354;>PTJWVU=HIsDT@?7:;>=HM\4TV ;>PTJ[VUJWI1M\9U6TPFM\@?M\ZA=?JKV XTJ[hAJK@HM\6CJ[X1hAJK9an B 7AV<; 7A4TX JK9<Jw7A@HVUMrVa; 75T@HJW7AZL7A=H4TV<;9>7A4TXFM\I _Q7ADFVUVU=?7A4 7TDTe[;UD87:;>=HM\4TVQ;>P87:; JK9UJpDFVUJKX ;UM;UJKV<;];>PTJV<; 7)5T=?@H=H;5n MAB ;>PTJ687:;U;UJK9<4TV7AZL7A=H4TV<;m6gJK9a;>DT9 587:;U=?M\4TVW7:VbXF=?VUe[DTVUV<JKXJ 7A9<@?=?J[9Kx

3M0

?_=?Z\DF9UJ%3Tx 3Tj _7:;U;UJK9<4TV|M5F; 7A=H4TJKXpV<=?IsDF@ 7:;U=?M\4MAB ;UPTJrZ\J[4TJK9U7A@`2ODT9U=H4TZ IMiXTJK@ 7:B ;>J[9500 000=H;>J[9>7:;U=?M\4FVKx J[B ; j354F=H;>=?7A@\e[M\4TXT= ;>=?MA4TVKje PTJ[I=He 7A@ V<J[; =?4QI=HX.B56T@?7A4TJ!52? =?4T= ;>= 7:@:e[M\4TXT= ;>=?MA4TVKj

e PTJKI1=?eK7A@ 6T@ 7:eKJKX M\4 0K0K0B56T@?7A4TJw=H47A4;U9U=?7A4TZ\DT@?7A9 I1JKV<P)x

> > = G AJM P T PWJ JPWGIT

354 MA9UXTJ[9Y;UMpV<;UDTXFn WPTJu;>PTJ[9Q= ;r=?Vr6gM\V<VU=65F@?Jo;UMM5F; 7A=H4 6TDT9<J1MA9rVU=?4FZ\@?JsI1MiXTJKV J VUJu;QDT6;>PTJV<niV<;UJKI VUDTe P ;UP87:; ;>PFJ"=?4F=H;>=?7A@ e[M\4TXT= ;>=HM\4TV VUPTMADT@?X B 7 hAM\91M\4T@Hn MA4TJV<JK@HJKe[;UJKX I1MiXTJAx?_=?Z\DF9UJ 3Tx 3L71V<PTMWVr7 VU=H;UD87:;U=?M\4 WPTJK9<J];UPTJ]VaniV<;>J[I 7AV|6T9UJ[687A9UJ[X`=?4VUDFe P7 7 n;UP87:;|7@ 7 n\J[9|MAB e PTJKI1=?eK7A@ 7AVwV<J[; MA4T@Hn =H4;>PFJoI1=?X.B 6T@ 7A4FJrMAB ;>PFJoVU=HIsDT@?7:;>=HM\4 5CM@;>M6T9UM hi=?XFJB 7 hAM\9>7)5T@?J =?4T= ;>= 7:@\eKM\4FXT=H;U=?M\4TVOBEM\9 ;>PTJN@?7AI1JK@?@?7A9V<;U9UDTeu;>DT9<J ;>MwXTJ[hAJK@?MA6)x[2mPTJN6T7A9>7AI1J[;UJK9<V 7A9<J;>PTJYV>7:IJQ7AVmDTV<JKX=?4H?_=?ZADT9UJ 3Tx 5)x 354 ?_=?ZADT9UJ%3Fx93K5e PTJ[I=He 7A@ 7AVb=?4L;>9<MfXTDFeKJKXM\4T@ n=H4@?Mie 7:;U=?M\4TVleKMA9U9UJ[VU6gM\4TXT=H4TZs;>M:0K0K0lB 6T@ 7A4FJ|=H4 7A4;>9<= 7A4TZADT@ 7A9 I1JKV<PT= x JAxEfB 7 hAM\9U=H4TZsPTJ @T7:Z\M\487A@V<niII1J[;U9<nMAB ;UPTJYXT9UMA6T@?Ju;>VmMK5F;>7A=?4FJKX7AVW=?4 ?_=HZ\DT9<J 3Tx :eAxd354H5CMA;UP e 7AV<JKVbe PTJKI1=?eK7A@c 7AV=?4L;>9<MfXTDFeKJKX"DT4T=HBEMA9UI1@HnpM h\J[9w;>PFJoeKD&5CJ 7:Vw=?4^?_=?ZADT9UJ3Tx Fx|kmeu;>D87:@?@Hn = ;wXF=?X`4TM:;r7)XCJKeu;w;>PFJ;>M\6gM\@?MAZAn"M:B ;UPTJ 9UJKV<DT@H;U=?4TZpVa;>9<DTe[;UDT9UJ[V WPF=?e P M\4TJ MABl;>PTJ I1M\9U6TPFM\Z\JK4FV 7AVY=?4L;U9UMiXTDTeKJ[X7AV<niI1IJu;>9U=He 7A@H@HnAx 2mPFJVa; 75F=?@?= ;5n M:B|;UPTJKV<J6T7:;U;UJK9U4FV 7AV17A@HVUM`;>J[V<;UJKX 7:ZL7A=?4FV<; _Y7ADTV<VU= 7:44TM\=?V<JY7A4TXH5CM:;>P MABO;>PTJ[I ;>DT9<4TJKXM\DF;m;UM 5gJY9UMK5FDTV<;Kx

?_=?Z\DF9UJ 3Tx<7YVUPTMWVW7QVU=?IoDT@ 7:;U=?M\4 9<JKVUDF@H;Ne[M\9U9<JKV<6CM\4FXT=?4TZ];>M ? =?Z\DT9<J 3Fx93L7]7AVm7AXFXT=H;U=Hh\J _Y7ADTV BVU= 7:4"4TM\=HVUJ N7:VwDTV<JKX =?4 ;>PFJ 7 nXTJ[VUeK9<=65gJKX`=H4;UPTJZ5gJKZA=?4T4T=H4TZ1MAB ;>PF=?VbVUJKeu;>=HM\4)x 4TJoeK7A4MK5TVUJ[9<hAJ ;UP87:; DFVU=?4FZ";>PTJp=H4T=H;U= 7A@de[M\4TXT= ;>=HM\4TVo=H4 ;>PFJpI=HXT6T@ 7:4TJ JpXT9<=Hh\J;>PTJpVanfVa;>J[I 9UM)B5TDTV<;U@Hn1;>Ms;UPTJ @ 7:IJ[@?@ 7:9lVa;>9<DTe[;UDT9UJ[VKx ydh\JK47sVUD&5TV<;>7A4L;>= 7:@)7AI1M\DT4L;mMAB 4FM\=?V<J XT=?Xp4FMA;O5T9<J 7AtXTMW4;>PFJQ6F@ 7A4TJ[V5TDf;|I7AXTJ ;UPTJKI 7:@?=?ZA4XT=1XCJK9<JK4L;>@ n 7A4TXeKDF9<h\J:xE2mPTJQVU6FPTJK9<=?e 7:@XT9UM\6F@?J[;V<;>9<DTe[;UDT9UJ[V J[9UJsIoDTe P IMA9UJQV<; 7)5T@?Jo7AZL7A=H4TV<;w7AXTXT= ;>= h\Jo4FM\=?V<J<C7A4TX7AXTXF=?4TZ4TM\=?V<JoXT=HX"4TM:;7)X J[e[;d;UPTJKI=?4 7A4Ln 4TM:;>=?e[J 75F@?J 7 n\x 2mPT=HVd=?V XFDTJm;>M ;>PFJW7AI6F@?=H;UDTXTJ MAB ;>PTJmDT4FV<; 7)5T@?J 7 h\Jh\JKeu;>M\9<V 7AV 7AVwXT=HVUe[DTVUV<JKX =?4â P87A6f;>JK9wzfx#AWMK5TDTVa;>4TJ[VUV|MAB ;>PFJ]I1MfXFJK@=?VWh\J[9<n=?I16gM\9<;>7A4L;

3Sz

?_=?Z\DF9UJ 3Txb7fj AWJKV<DT@H;UVrMAB ;>PTJ V<=?IsDT@?7:;>=HM\4 WPF=?e P N7:VQ=H4T=H;U= 7A@H=6U[JKX=?4`;>PFJ1VU7AI1JsI7:4T4TJK9Y7:V?_=?ZTx)3Tx 3L7 (5TDF; 7AXFXT=H;U=Hh\J _Y7ADTV<VU=?7A4 9>7A4TXFM\I&4TMA=?VUJ 7AVlDTVUJ[X)x 2mPTJW4TM\=HVUJW9UJ[VUDT@ ;>V =?4 e[DT9<hAJKX7A4TX V<;U9>7A4TZAJK@Hnp7A@?=HZ\4TJKXp6F@ 7A4TJ[VKx

VU=?4FeKJ WPTJK4 ;UPTJI1MiXTJK@HV]7:9UJ1DTVUJ[X BEM\9QV<;>DFXT=?J[V]MABO5T=?M\@HM\Z\=He 7A@ V<niV<;UJKI1V JP87 hAJ;>M ; 7At:J=?4L;>M17Ae[eKM\DT4L;b4TM\=HVUJY7A4FXMA;UPTJK9WXT=HV<;UDT9 5T=H4TZsB 7Aeu;>M\9<VW;UP87:;WJ @F=HV<;b=?4;UPTJY487:;UDT9UJ:x

> > DAR M@G T PQJ J PWGIT

4|J@i; J N7A4L;m;UM J@f6T@?MA9UJ ;>PFJrJlXCJKe[;WM:B IoDT@H;U=?6T@HJ|I1MfXFJKV ;UM ;UPTJrI1M\9U6TPFM\@?M\Z:n MAB ;>PTJrVanfVCB;>JKIxO] JoXT=HX;UPT=?V 5Ln;UDT4T=H4TZ;UPTJ]6T7A9>7AI1J[;UJK9<V|=H4"VUDFe P7 7 n ;UP87:;|IMiXTJ

k = 0.96N7AV

;>PTJbIMAV<;lDT4TVa; 75T@HJWM\4TJAxd354 ;>PFJ|V<=?IsDT@?7:;>=HM\4TV JwDFVUJKX687A9U7AIJu;>J[9UVα = 0.89

β = −0.99

δ = 0.25, D = 0.516

x 2mPTJY4TM\4T@H=?4TJK7A9N687A9U7AI1J[;>J[9UV JK9UJr1 = 0.02

7A4FXr2 = 0.2

eKM\9<9UJ[V B6CM\4FXT=?4TZp;UMXT9<M\6T@HJ[;Q687:;<;>JK9<4TVKxe4|MA;U=?eKJs;UP87:;]4TM;>PFJ1XF=?VU6gJK9<VU=HM\4 9<JK@?7:;>=HM\4 = ?_=HZ\DT9<J 3Fx10!?7A@?@HM WVbI7A4Ln"I1MfXFJKVw;UMZ\9UM = ;UPTJK9<Js=?V 7@?M\4FZVUJ[Z\IJ[4L;rM\4`;>PFJstKB 7@f=?VbBEM\9wDF4TV<;>75T@?JoJK=YBZ\JK4Lh:7A@?DFJKV

λ? 7A4TX ;>PT=HVWe 7ADTV<JKV|9U=?ZAM\9UM\DFVNeKM\I16gJ[;>= ;>=HM\4 7A4TX7A@?I1M\Va;me P87AMA;U=?eYI1M\9U6TPFM\@?M\Z:n\x

?_=?Z\DF9UJ 3Tx98\7bV<PTMWV_;>PFJ Va; 75T=H@?=1UKJKXQeKMA4FG8Z\DT9U7:;>=HM\4 WPTJK9<J ;>PFJ 687 ;U;>J[9U4sV<PTMWV 7W;>DM5TDF@ 7A9CB5@?=HtAJV<;>9<DTe[;UDT9UJ:x

354 MA9UXTJ[9Y;UMpMK5TV<JK9<hAJ1;UPTJ eKM\I16gJ[;>= ;>=HM\4 5gJ[; JKJK4 VU6TPFJK9U=He 7A@ 7A4TX @ 7AI1JK@H@ 7A9WV<;U9UDTeu;>DT9<JKV J=?VUMA@ 7:;UJKXI1MiXTJ

k = 0.457A4TXpV<J[;f5gMA;UP

r17A4FX

r2XT=1XCJK9<JK4L; BE9<M\I UKJK9<MTx|V<=?4TZo4TM\4.B @?=?4FJ 7A9

687A9>7:IJu;>JK9<Vr1 = 3.5

7A4TXr2 = 0.2

9<JKVUDF@H;>J[X =?4 ;>PTJe[M\I16CJu;>=H;U=?M\4 5gJ[; JKJK4 e[DM5T=?e7A4TXRiDT7AXT9>7 ;>=?er;UJK9UI1VmB 7 h\M\9<=?4TZ @ 7:IJ[@?@ 7:9N7A4TXVU6FPTJK9<=?e 7:@OVa;>9<DTe[;UDT9UJ[Vg9UJKV<6CJ[e[;U=Hh\J[@HnAxE2mPTJQ9UJ[VUDT@ ;MABW;>PT=HVseKM\I16gJ[;>= ;>=HM\4 eK7A4 5CJpV<JKJ[4 =H4 ?_=?ZADT9UJ 3Txb8)5 WPTJK9<J M\4TJeK7A4 eK@?JK7A9U@ nG84TX 9UM\DF4TX@ 7AI1JK@H@ 7A9WV<;U9UDTeu;>DT9<JKV W=H;UP VU6TPFJK9U=He 7A@_PTM\@?J[VKxY354 ;>PT=HV e 7:VUJ< = ;w;>MiM\t

2 000 000;U=?I1JoVa;>JK6FV

;>M V<;>75T=?@H=6U[Jw;>PTJYVa;>9UDFe[;>DF9UJAx3L

?_=?Z\DF9UJ 3Txb8fj> 7:;<;>J[9U4TVsM5F; 7A=H4TJKX DTVU=H4TZ";UPTJZ\J[4TJK9U7A@ 2 DT9<=?4TZV<niV<;UJKIx`7 ? k|B ;UJK9500 000;>=?I1JNVa;>JK6FVf;UPTJW=?V<M\@ 7:;UJKX I1MfXFJ N7AV

k = 0.967A4FX4TMA4.B5@H=?4TJK7A9_687:9>7AI1J[;UJK9UV

r1 = 0.027A4TX

r2 = 0.2 7A4FXH5@?N7 B ;>JK9

2 000 000;U=?I1J V<;UJK6TV

k = 0.45r1 = 3.5

7A4TXr2 = 0.2

x

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S(~k, t) =

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|S(k

)|²

?_=?Z\DF9UJ 3Tx9SFjlci;U9UDTeu;>DT9<JQB 7Aeu;>M\9<VmMAB_;UPTJY@ 7AI1JK@H@ 7A9 7A4TXVU6FPTJK9<=?e 7:@V<;>9<DTe[;UDT9UJ[V W= ;>Pk = 0.84= ?_=HZ\VKx 3Tx 527A4TX2X@? x 2mPTJV<;>9<DTe[;UDT9UJB 7Ae[;UM\9UV JK9UJ"7 h\JK9U7AZ\J[X M h\JK9\ Va; 75F@?J V>7AI16T@HJKVKx

4|MA;U=?eKJ1;>P87 ;s;>PTJ6CJK7AtSV =?4 7`7:4TX 5 7A9UJ=?4 ;UPTJV>7:IJ6CM\V<=H;U=?M\4 =?4 ;UPTJptKB 7@f=?V = x JAxE ;>PFJe P87A9>7:e[;>J[9U=HV<;>=He 7 h\J @HJK4TZA;UP"=HVb;UPTJoVU7AIJQ=?4 5CM:;>P"Va;>9<DTe[;UDT9UJ[Vr7AVw= ;bVUPTM\DF@?X 587:VUJKX"M\4";>PFJ@?=?4FJ 7A9m7:487A@HniV<=?VKx

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

k

|S(k

)|²

?_=?Z\DF9UJ 3Tx9TFj ci;>9<DTe[;UDT9UJrB 7Aeu;>M\9WMABO;>PFJ]e[M\I6gJ[;U=H;U=?M\4M:B VU6FPTJK9<=?e 7:@ 7A4TX@ 7AI1JK@H@ 7A9lV<;U9UDTeu;>DT9<JKVW=H;>P

k = 0.45= ?_=?ZFx3Tx9852?>x 2mPTJ 6T@?MA; eKM\I 5F=?4TJ[VQRSD87:4S;U=H;>7:;>= h\JK@ n;UPTJ e P87A9U7Ae[;UJK9U=HV<;U=?eKVYMAB

;>PTJr; MeK7AVUJ[V|=H4 ? =?Z\DT9<J%3Tx9SFx

3(S

%2

"2.1 $ ' $ -

2mPT=?V]e P87A6F;UJK9 eKMA4TeKJ[4S;U9>7:;UJKV M\4 7"IMA9UJe[M\I16T@?=He 7:;UJKX 9UJ 7:e[;>=HM\4.B XT=1XCDTVU=HM\4 V<niV<;UJKI WPT=He PXTJKV<eK9U=15CJ[VY=?9U9<J[hAJK9UV<=65T@HJse PTJ[I=He 7A@ 9UJ 7:e[;>=HM\4TV[x ?_=H9UV<; J1V<PTMPTM9UJK7Ae[;U=?M\4 tS=?4FJ[;>=HeKVreK7A45CJ XTJu;>J[9UI1=?4TJ[X BE9UM\I BEM\9UIsDF@ 7AJMABQe PTJKI1=?eK7A@ 9UJK7Ae[;U=?M\4FVKx ] J W=?@H@|DFVUJ@?=H4TJ 7A97A487A@ niVU=?V;>M6T9UJ[XT=?eu;Q;UPTJ 5CJ[P87 hf=HM\9YMAB ;UPTJ1V<niV<;UJKI=?4 ;>PFJVU7AIJ I7A4T4TJ[9Q7AV]=H46T9UJuhf=HM\DTVQe P87A6F;UJK9[x] PTJK4 V<M\DT9Ue[JKVdMABge PTJ[I=He 7A@HV_7:9UJm7AXFXTJKXs;UMw;>PTJ V<niV<;UJKI Jme 7A4 M5TVUJ[9<h\JmXFJK4TXT9<=H;U=?e ZA9UMm;>P)x?_=?487:@?@Hn< J W=?@?@S6T9<M\6gM\VUJ 7 5T=?MA@?M\Z\=He 7A@:IMiXTJ[@ WPT=?e P =HV587:VUJKX MA4];>PFJ#_Y9U7 nB cfeKM:;U; V<niV<;UJKIx

&(' )4* , 63;*

2mPTJ VUJ[eKM\4FXVaniV<;>J[I J P87 h\JrVa;>DTXT=HJKXp=HV ;>PFJ _Y9U7 nB cfeKM:;U; 9_r9>7 n7:4TX cfe[MA;U;0TKSA! I1MiXTJK@;>P87:; 7AV`6T9<M\6CMAVUJKX ;>M2XFJKVUe[9U=15CJ ; M =H9U9UJuh\J[9UVU=15T@?J 7ADF;UMfeK7:; 7:@HnS;>=He e PFJKI1=?e 7:@Q9<J 7Aeu;>=HM\4TVKxkwDF;UMfeK7:; 7A@ niVU=?VI1J 7:4TVO;UP87:; ;>PFJ 6T9<MK5875F=?@?= ;5nwM:BTMieKeKDF9U9UJ[4TeKJ MABTV<M\I1JlI1M\9<6TPTM\@HM\ZAn MA9Oe PFJKI B=?e 7:@8=?4Te[9UJ 7:VUJKV 7AV 7QBEDT4Teu;>=HM\4M:B);>PTJb4SDTI 5gJK9 MAB)V<DTe PpIMA9U6TPTMA@?M\Z\=HJKV MA9l;UPTJ|e[M\4Te[JK4L;>9U7:;>=HM\4MAB_;UPTJoe PTJ[I=He 7A@O7A@H9UJK7AXFnp6T9<JKV<JK4L;rV<niVa;>JKIx#2mPTJo9<J 7Aeu;>=HM\4 J 7A4L;w;UMXTJ[VUe[9U=65gJo=HVWMAB ;>PFJBEM\@?@HM W=H4TZoBEM\9PTJ 9<J 7Aeu;>=HM\4`MABlM\4TJ DF4T=H;rM:B e PTJKI1=?eK7A@ 7A4FX`; MDT4T= ;>V MAB c6F9UMiXTDTeKJ[VQ7ADF;>Mie 7 ; 7B@HnS;>=He 7A@H@Hn;UPT9UJ[J DT4T=H;UV MAB csxCv DTJ ;>M;>PTJ =H9U9UJuh\J[9UVU=15T@?Jo487:;>DF9UJsMAB ;UPTJs9UJK7Ae[;U=?M\4FV)e PTJ[I=He 7A@=?VY7:4 =H4TJK9a;r6T9<MfXTDFe[; xo354 ;UPTJ 9UJK7Ae[;U=?M\4 BEM\9<IsDT@ 7687A9>7:IJu;>JK9<V

r17A4TX

r2V<;>7A4TX BEM\9 ;>PFJ

3(T

9UJ 7:e[;>=HM\4 9U7:;>Je[M\4TVa; 7A4L;>V[x 354 M\9<XTJK9s;UM W9U= ;>JXFM W4 ;UPTJpXT=1XCJK9<JK4L;>=?7A@lJKRSD87:;U=?M\4TV]BEM\9o;>PFJ6T9UMieKJ[VUV= ;w=HVr7AV<VUDTI1JKX`;UP87:; ;UPTJsV<niVa;>JKI =?Vw=H4èKMA4S;>7Ae[; W=H;>P79UJKV<JK9ah\M\=H9 MABle PTJKI1=?eK7A@W=H;>P7]e[M\4TV<;>7A4L;N9U7:;>J

F7:4TX;UP87:;Ne PFJKI1=?e 7:@!c =?Vl9UJ[IM hAJKXBE9UMAI;>PTJ|V<niV<;UJKI 5Sn1;>PFJbBEJ[JKX

6T9UMieKJ[VUVb7A4TX=?4 7AXFXT=H;U=?M\4pe PTJ[I=He 7A@jc =?VmeKMA4ShAJK9a;>JKX;>M4 W= ;>P 9>7:;UJYq xM] PTJK4 JYe[M\@?@HJKe[;;>PT=HV J]M5F; 7A=H4p9UJK7Ae[;U=?M\4TV

+ 2

c →1 3c

c →K

c →F

F

.

= 7ix z ?

kmeKeKMA9UXT=H4TZo;>MQ;>PTJ|@ 7 MAB I7AV<V 7Aeu;>=HM\4 = JAx ZTx ^DT9U9U7 n@0TKT\6 ?f=H4e PTJ[I=He 7A@ 9UJK7Ae[;U=?M\4FV ;>PFJ9>7:;UJ1M:BN9UJK7Ae[;U=?M\4 =?VY6T9<M\6gM\9<;U=?M\487:@ ;>M;>PTJ 6T9<MfXFDTe[;]MABNe[M\4Te[JK4L;>9U7:;>=HM\4TVQMABN9<J 7Aeu; 7A4L;>V[x:] JVUP87A@H@ XTJ[4TMA;UJ ;UPTJpeKMA4TeKJ[4S;U9>7:;U=?M\4FV 5Ln

u = [U ]7:4TX

v = [V ]x-4|M J e 7A4 W9<=H;>J;UPTJ

9UJ 7:e[;>=HM\4 tS=?4TJu;>=?e[VNBEM\9N;>PFJQe PFJKI1=?e 7:@?Vu7:4TX

v7AV

f(u, v) = −uv2 + F − Fu = −uv2 + F (1 − u)

g(u, v) = −2uv2 + 3uv2 − Fv − Kv = uv2 − v(F + K).= 7fx ?

2mPTJwJ[RiDT7:;>=HM\4TVlMAB)I1MA;>=HM\4 BEM\9d;>PTJ|eKM\4FeKJK4L;U9>7:;U=?M\4TVlMABC;>PTJb; M]e PTJKI1=?eK7A@?Vu ≡ u(~x, t)

7A4TXv ≡ v(~x, t)

=H4XT=HIJ[4TVU=HM\4T@HJKVUV DT4F=H;>V e 7A4:5CJ|MK5F; 7:=?4TJ[X:5Ln1VUDM5TVa;>= ;>DF;U=?4TZ]ylR x;= 7fx ?_=?4L;>MQ;>PTJZ\JK4TJ[9>7A@)BEMA9UI M:B\2ODT9U=H4TZ V<niV<;UJKI = y R x8= 3Fx10!?'? x 2mPT=HVmM\6gJK9>7 ;>=?MA4 ni=HJK@?XFV

ut = Du∇2u − uv2 + F (1 − u)

vt = Dv∇2v + uv2 − (F + K)v,= 7fx 3?

WPTJK9<JQ;UPTJYXT=1XCDTVU=HM\4peKMiJ1e[=?JK4L;UVbBEM\9N;UPTJ ; Me PTJ[I=He 7A@HVm7A9UJDu

7A4TXDv

89<JKVU6gJKeu;>= h\JK@ n\x

2mPTJ_Y9U7 nB cfeKM:;U;WI1MfXTJ[@ 7AVmV<;UDTXT=HJKX7A487A@ nS;>=?eK7A@?@ n17A4TX 4SDTI1JK9U=He 7A@H@Hn1=?4; M XT=?I1JK4FVU=?MA4TV5Sn J 7A9<VUM\4 Y0TT\ V<JKJ 7:@?VUM PL;U;U6)j Yx e 7Ae[9Kx e 7A@ ;>JKe P x J[XTD :=HVUI7A6 :=HI7AZAJAx PS;UI@E WPTMI7A6F6CJ[X";>PTJ]6TP87AV<JsXT=?7AZ\9>7:I MABd;>PTJ]V<niV<;UJKI%=?4;UJK9<IV|MABd;>PTJo9>7:;UJoeKMA4TV<;>7A4L;>V

F7A4TX

Kx

2mPT=?V I1MfXTJ[@LJ@fPT=15T=H;UV 7mh\J[9<nQ9U=He PZ5gJKPT7 hi=HM\9 9>7A4TZA=?4TZmBE9UMAI;>=HIJ B5=?4FXTJK6gJK4TXFJK4L;OV<;>JK7AXFn(B5Va; 7:;UJVUM\@HDF;>=HM\4TVY;>M"e PT7AMA;>=He< MAVUeK=H@?@?7:;>M\9an 7A4TX ;UM;U=?I1JB5XFJK6gJK4TXTJ[4L;s6TP87:VUJ;UDT9 5TDF@?JK4L; 5CJ[P87 hf=HM\9Kx? DF9<;>PFJK9UI1M\9<J< cl7AV<;>7A4TMpJu;Q7A@ xE0TKS(;6_P87 h\J VUPTMW4 ;>PT7:; ;>PFJ V<niV<;UJKI XTJuh\JK@HM\6TVrV<687:;>=?7A@?@ nV<;>JK7AXFn687 ;U;>J[9U4TVQJ[hAJK4 WPFJK4 ;UPTJ XT=1XCDTV<=?M\4eKMA4TV<;>7A4L;>VoM:B ;UPTJ1; M"e PFJKI1=?e 7:@?VY7A9<J1J[RSD87A@ x2mPT=?V 5gJKP87 hi=?MA9d=?Vd687A9a;>=HeKDT@?7A9 ;UMY;UPTJWM\4TJ B5XT=HIJ[4TVU=HM\487A@fe 7AV<J|7:4TX =H;dP87:Vl4FMA; 5CJ[JK41MK5TV<JK9<hAJKX=?4 PT=?ZAPTJK9mXT=HIJ[4TVU=HM\487A@CV<niV<;UJKI1VKx

7A

3542;>PT=HVVa;>DTXfn JG89UVa;6gJK9<BEMA9UI1JKX24SDTI1JK9<=?eK7A@|V<=?IsDF@ 7:;U=?M\4TV M:B ;>PTJ _Y9U7 nB cfeKM:;U;I1MiXTJK@=?4 ;>PF9UJKJ XT=HIJ[4TVU=HM\4TV[x k|B ;UJK9s;>PT7:; J 9U7A4TXTM\I1@Hn 7AXTXFJKX 7V<I7:@?@l4iDFI 5CJ[9 VUM\DT9<eKJ[V1BEM\9e PTJKI1=?eK7A@&csx(2mPFJ|I1MA;>= h:7:;>=HM\4]BEM\9 7:XTXT=?4FZY;UPTJWVUM\DF9UeKJ[V 7AVd;>MQ=?4Lh\J[V<;U=?ZL7:;UJN;>PTJW9<MK5TDTVa;>4TJ[VUVMAB 687 ;U;>J[9U4TV7A4TX ;>PTJ BEM\9<I7:;U=?M\4 M:B eKMA4T4TJKeu;>=HM\4TV:5gJ[; JKJ[42;>PFJV<M\DT9Ue[JKV=?4 7A4 7:;<;>J[I6F;;>M"I1=?I1=?e];>PTJXTJ[hAJK@?MA6TIJ[4L;sMABW7H5T=?MA@?M\Z\=He 7A@_4TJ[; M\9UtdJ:x ZTxE ;UPTJBEMA9UI7:;>=HM\4MABmV<ni487:6F;>=HeeKM\4L; 7:e[;>V 5gJ[; JKJ[4 4TJKDF9UM\4TV[x |M Juh\J[9_= ;r=?V nAJ[;QDT4Te[@?JK7A9 WPTJ[;UPTJK9 2ODT9U=H4TZ V<niV<;UJKI1V]eK7A46T9UMiXTDTe[J VUDTe P 7A4 =?4FXTDTe[;U=HhAJsVU=HZ\487A@H=?4TZIJ[e P87A4T=HVUI BEM\9 4TJ[DT9>7A@_687:;U;UJK9<4T=?4TZ:5TDF;r;>PFJ =?XTJK7=?Vm4TJuh\JK9a;>PTJ[@?J[VUV 7:6T6CJK7A@?=H4TZTx

& % 1;* /+ /214/ . .

kwV N7AV_7A@?9<J 7AXfnYXT=?V<eKDTV<VUJKXo=?4Y;>PTJ 6T9UJuhf=HM\DTVOe P87A6F;UJK9A;>PTJl6TDT9<6CMAVUJlMABT;>PFJ @H=?4TJK7A9 7A487A@ niVU=?VO=?V;>MY6T9UJ[XT=?eu;d;>PTJWVa; 75T=H@?= ;5no7:4TX e P87A9>7:e[;>J[9U=HV<;>=HeKV MAB ;>PFJbZA9UMm;>P1=?4 7rV<niVa;>JKIx 2ODT9U=H4TZrVUPTM JKX;>P87:;QDT4FXTJK9]eKJK9a; 7A=H4 e[M\4TXT= ;>=?MA4TV BEM\9Y;>PTJ1687A9U7AIJu;>J[9UV W=H;UPTM\DF;]XT=YX DTV<=?M\4 7V<;>JK7AXFn(B5Va; 7:;UJeKM\DT@HX^5gJs@?=H4TJ 7:9U@HnV<; 7)5T@?J!j5TDf; =?4;>PTJo6T9UJ[VUJK4FeKJ MAB XT=YX DFVU=?MA4DT4FV<; 7)5T@?J:x 2mPTJs6T7A9>7AI1J[;UJK9<VMAB_;UPTJZ2ODT9<=?4TZ1V<niV<;UJKI VU6T7A4`7 687A9U7AIJu;>J[9bV<687Ae[Js7A@HVUM tS4TMW4 7AV#2 DF9U=?4FZV<687AeKJ:xm354V<M\I1JVUDM5gXTM\I7A=?4 MABm;>PF=?VoV<687Ae[J JpeK7A4 eK9<J 7:;UJpVUDT= ; 75F@?J1eK=?9<eKDTI1V<;>7A4Te[JKVoBEM\9]XT=1XCDTV<=?M\4.B XT9U= h\J[4=?4TVa; 75T=H@?= ;5n\x DT9mVaniV<;>J[I =?VmZ\= h\JK47AV

ut = Du∆u − uv2 + F (1 − u)

vt = Dv∆v + uv2 − (F + K)v.

= 7ix<7 ?

_J 7:9UVUMA4 0TKTA mVUJ[J7A@?V<M PL;U;>6 j Yx eK7AeK9[x eK7A@H;UJKe P)x JKXTD :=?V<I7A6 :=?I7AZ\J:x PL;>I1@BbP87:Vp7A487B@HnVUKJ[X ;>PT=HVsIMiXTJ[@N7A4TX I7A6T6gJKX ;>PTJp6TPT7AVUJpXT=?7AZ\9U7AIMABW;>PTJVaniV<;>J[IxI= n VUJ[;<;>=H4TZ`;>=HIJhA7:9U= 7 ;>=?MA4p7A4TX ;>PTJYXT=YX DFVU=?MA4;>JK9<IVN;UM U[JK9<M7:4TXV<M\@Hhi=H4TZs;>PTJYJ[RiDT7:;>=HM\4TV JQZ\Ju;

v0 =F (1 − u0)

F + K.

= 7ixb8 ?

?_=Y@f=?4FZu0 = 1

ni=?J[@?XTV ;>PFJVa;>J 7:XFn(B5V<;>7:;>JVUM\@HDF;>=HM\4(u0, v0) = (1, 0)

x+=ln V<DM5TVa;>=H;UDF;>=H4TZylR x = 7ixb8 ?5T7Ae t ;>M ;>PFJ9<J 7Aeu;>=HM\4 ;UJK9<IV MABwy Rgx= 7fxb7 ? J MK5F;>7A=?4 7A4FMA;>PTJ[9 V<;>JK7AXFn(B5Va; 7:;UJVUM\@HDF;>=HM\4pXTJ[GT4TJKX 5SnylR x = 7ixb8 ? 7:4TX

u0 =1 ±

√

1 − 4(F+K)2

F

2.

= 7ix<; ?

7.0

= no;UPTJmV>7AI1JmtS=?4TX MAB ;U9UJ 7 ;>I1JK4L;l7AV =H4s;>PTJW6T9<J[hi=?M\DFVdVUJKeu;>=HM\417A4TX 5LnsDTV<=?4TZr;>9<= 7A@iVUMA@?DF;U=?M\4TVXTJ[G84FJKX^5Lnpy RSVKx =J3Txb8 ? 7A4TX = 3Txb; ? JQeK7A4 W9U=H;UJry Rgx = 7fx<70?l=?4p;>PTJY@H=?4TJK7A9U=1UKJKXBEM\9UI

wt = D∆w + Aw,= 7fx9S ?

WPTJK9<Jw =

(

u − u0

v − v0

)

,

D =

(

Du 0

0 Dv

)

,

A =

(

fu fv

gu gv

)

,

W=H;>Pp;UPTJrBEM\@?@HM W=H4TZsV<PTM\9<; B5P87:4TX 4TMA; 7 ;>=?MA4TV

fu = −v20 − F

= 7fxbT ?fv = −2u0v0

= 7fxY0 ?gu = v2

0

= 7fxY0K0!?gv = 2u0v0 − (F + K).

= 7fx10 z ?

4|M;UPTJYXT=?V<6CJ[9UVU=HM\4 9UJ[@ 7:;U=?M\4=HVWVUM\@ h\JKX BE9UMAI

|λI − A + Dk2| = 0,= 7ix10 ?

WPT=?e PV<;>7A4TXTVmBEM\9∣

∣

∣

∣

∣

λ − fu + Duk2 −fv

−gu λ − gv + Dvk2

∣

∣

∣

∣

∣

= 0.

? MA9|;>PFJoV<;UJ 7AXfnB V<;>7:;>J V<M\@?DF;U=?M\4(u0, v0) = (1, 0)

Jse 7:4 J 7AV<=?@HnV<M\@HhAJo;>PTJ]e P87A9U7Ae[;UJK9U=HV<;U=?e6CM\@ ni4TM\I1= 7A@j5Sn eK7A@?e[DT@ 7 ;>=?4FZs;>PTJQXTJ[;UJK9<I=H487A4L; x 2mPTJ U[JK9UMAV|MAB ;>PTJQe P87A9U7Ae[;UJK9<=?V<;U=?eY6gM\@Hni4TM)BI=?7A@ i= x JAxEF;>PFJQJ[=?Z\J[4LhA7:@?DTJ[VW7A9UJYZ\= h\JK47AV

λ1 = −Duk2 − F, λ2 = −Dvk

2 − F − K.= 7ix103 ?

= MA;>P ;>PTJ`J[=?Z\J[4LhA7:@?DTJ[V7A9<J`4TJKZ\7:;>= h\J< WPT=He P 6T9UJ[XT=?eu;>V=?42;UPTJ`VUJ[4TVUJ M:BY@?=?4FJ 7A97A487A@ niVU=?V;>P87:; ;>PTJrVa;>J 7:XFn(B5V<;>7:;>J

(1, 0)XTMiJKVm4TMA;#5F=HBEDT9<e 7:;UJ<F= x J:xBf;>PTJ[9UJ W=?@H@5gJr4TJK= ;>PTJ[9NZ\9<M m;UP 4TM\9

e P87A9>7:e[;>J[9U=HV<;>=HeY@?JK4FZA;>PVUeK7A@?JQ=?4 ;>PTJQV<niVa;>JKIxλ1(k)

=HVW6T@?MA;<;>J[X=H4^?_=HZ\DT9<J7fx10 = V<M\@?=HX @?=?4FJ6?/WPTJK9<JbMA4TJme 7A4 4TM:;>=?e[J ;>PT7:;

λ1(k)XTMfJ[Vd4TMA; 5gM\DT4TX17A4Lno6CM\V<=H;U=Hh\J IMiXTJKV 7:V ;>PTJN; MYMA;>PTJ[9

XT=?V<6CJ[9UVU=HM\4 9UJ[@ 7:;U=?M\4TV =?4H?_=?ZADT9UJ 7ix10\x

7\z

? MA9l;UPTJ|4FM\4L;>9U= hi= 7A@ V<;UJ 7AXfnB V<;>7:;>J|XTJ[G84FJKX 5Ln1ylRSV[x= 7ixb8 ? 7A4TX = 7fxb; ? ;>PTJ|e 7A@HeKDT@?7:;>=HM\4 MAB);>PFJXTJ[;UJK9UI1=?4T7A4L;mnf=HJK@HXTVN;>PTJrBEM\@H@?MW=?4FZse P87A9U7Ae[;UJK9U=HV<;U=?er6gM\@Hni4TM\I1= 7:@

λ2 + λ(v20 − K + k2(Du + Dv)) + h(k2),

= 7fxY07 ?

WPTJK9<J

h(k2) = DuDvk4 + (Dv(v

20 + F ) − Du(F + K))k2 + (F + K)(v2

0 − F ).= 7ix108 ?

v DTJ;UM";>PTJ1BEM\9UIMABm;UPTJe P87:9>7Aeu;>JK9<=?Va;>=?e1JKRSD87 ;>=?MA4 =H;]=?V]P87A9<X ;>M`V<M\@HhAJp7A4TX JeK7A4T4TM:;BEM\@?@HM ;UPTJr6T9<Mfe[JKXTDT9<J JYDTV<JKX =?4;UPTJrV<;UJ 7AXfnB V<;>7:;>JYe 7:VUJ

(u0, v0) = (1, 0)x bM Juh\JK9 J

e 7A4J[h:7A@HD87:;>Jr;UPTJYeK9U= ;>=He 7A@ 7 h\JQh\JKeu;>M\9 5Ln 4TMA;U=?eK=H4TZ ;>P87 ;m;>PTJQ9UJ[Z\=?M\4 W= ;>P6CM\V<=H;U=Hh\Jλ(k)=?4 6TP87AV<JVU6T7AeKJ=?V5CM\DF4TXTJKX-5Sn;UPTJ1;UPTJ6gM\=H4S;UVYZ\=HhAJK4-5Ln UKJ[9UM\V]MAB

h(k2)x = J[e 7ADTV<J J

N7A4L;|;>M1=?VUMA@ 7:;UJY7AVbBEJ IMiXTJ[V|7:V|6gM\VUV<=65T@HJ JQeKM\4Te[JK4L;>9U7:;>J];>M ;>PTJQe 7AV<Jo=H4 WPT=He Ph(k2)P87AVWM\4T@ nM\4TJ U[JK9UMFx 2mPT=?VWMieKeKDF9UV WPTJK4;>PTJYXT=HVUe[9U=?I1=?4T7A4L;N=?VfU[JK9<M58= x JAxE

(Dv(v20 + F ) − Du(F + K))2 − 4DuDv(F + K)(v2

0 − F ) = 0.= 7fx10; ?

3 B_;>PF=?VW9UJ[@ 7:;U=?M\4p=HVbVU7:;>=HV<G8J[X8;>PTJQVUMA@?DF;U=?M\4 MAB y Rgx = 7ix108 ? 7A4TX;UPTJYeK9<=H;>=He 7A@N7 h\JQh\J[e[;>MA9w=?VZ\=HhAJK4H5Sn

k2c =

Du(F + K) − Dv(v20 − F )

2DuDv

.= 7fxY0S ?

2mPTJ`XT=HVU6gJK9UV<=?M\4 9UJ[@ 7:;U=?M\4TV =H@?@?DFV<;>9U7:;>=H4TZ ;>PTJ W=?4TXFM M:B N7 hAJ`h\JKeu;>M\9<VBEM\91;>PTJZ\9UMW=?4FZIMiXTJKV MABb;>PFJ; M V<JK@?J[e[;UJKX26FP87AVUJ[V7:9UJ 6T9UJ[VUJ[4S;UJKX =H4i? =?Z\DT9<JH7fxY0\xi2mPFJe[9U=H;U=?eK7A@ 7 h\Jh\JKeu;>M\9mXFJ[G84TJ[X 5Lnpy R x = 7fx10S ? Z\= h\JKVN;UPTJ

kh:7A@HDTJ eKMA9U9UJ[VU6gM\4TXT=H4TZ ;>Ms;UPTJrI7@F=HIsDTIh:7A@?DFJ

MABλ(k)

T= x J:xBf;>PTJrI1M\V<;WDF4TV<;>75T@?JrZA9UMm;>PIMiXTJ:x 4TJYVUPTM\DF@?X 4TMA;U=?eKJ ;UP87:;m=H4p;>PTJYe 7:VUJYMAB;>PTJ _r9>7 n(BacieKMA;<;NI1MfXTJ[@8;>PTJ 4SDTI 5CJ[9NMABO7A@H@?M J[XI1MfXFJKVN=HV @?7A9UZ\J[9 ;UP87A4=H4;>PTJwe 7AV<J MAB;>PFJZ\JK4TJ[9>7A@2ODT9<=?4TZ V<niVa;>JKIx

& , / : (671 + * . (:<.

354`;>PT=HVwVUJ[e[;U=?M\4 J W=?@?@ V<PTM;UP87:;w;UPTJ _r9>7 n(Bacfe[MA;U;wIMiXTJ[@ 6T9<MfXFDTeKJ[V eKM\I16T@HJ@;>DM5FDT@ 7A9V<;>9<DTe[;UDT9UJ[V WPTJK4 V<M\DT9<eKJKV M:BWIM\9<6TPTM\ZAJK4 7A9<Jp7AXTXFJKX ;>M;>PTJMA9U=?ZA=?487A@dVaniV<;>J[Ix ] J W=H@?@7A@?V<M1XF=?VUe[DTVUV|=H;UV 5F=?M\@HM\Z\=?eK7A@g=?I16T@?=He 7:;U=?M\4TVlBEM\9WIMiXTJ[@?=?4FZ 4TJKDT9<M\487A@ Z\9<Mm;>P=?4IM\9<JYXTJ[;>7A=?@7A4TX VUPTMV<M\I1JQ6F9UJK@H=?I1=?487:9<n 9UJ[VUDT@ ;>V[x

7A

0 0.5 1 1.5 2−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

λ (k

)

wave number k

?_=?Z\DF9UJ 7fx10Ajf2mPTJsXT=HVU6gJK9UV<=?M\49UJ[@ 7:;U=?M\4MAB ;UPTJ _r9>7 n(Bacfe[MA;U;|IMiXTJ[@OBEM\9Du = 0.125

Dv =

0.05F = 0.045

K = 0.065

4TJK7A9];UPTJVa;>J 7:XFn(B5V<;>7:;>JVUM\@HDF;>=HM\4(u0, v0) = (1, 0)

= V<M\@?=HX@?=?4FJ6?/A7A4TX

F = 0.065K = 0.0625

= X87AVUPFJKX @?=H4TJ6?)7A4TXF = 0.045

K = 0.065

= X87AV<P.B5XFMA;@?=?4FJ6?|4TJ 7:9];>PFJ4TM\4L;U9U=Hhi=?7A@dV<;UJ 7AXFn(B V<; 7 ;>JV<M\@?DF;U=?M\4 x 2mPTJp7A9<J 7"7)5CM h\J

kB57@f=?V5CM\DF4TXTVo;UPTJ

Z\9UMW=?4FZ1J[=?Z\J[4TI1MfXTJ[VKx

7)3

9> >B? GIF PWK K PWD

|VU=H4TZ;UPTJ _Y9U7 nB cfeKM:;U; I1MiXTJK@ MAB y Rgx = 7fx 3? J 6CJ[9<BEM\9<IJ[X";>PFJ VU=?IoDT@ 7:;U=?M\4FVb=H4"; M7:4TX;>PT9<JKJ"XF=?I1JK4TV<=?M\4TV[x = n2V<e 7A4F4T=?4TZ ;>PTJ6TP87AV<J`VU6T7AeKJ J`MK5f; 7A=H4TJKX h\JK9an29<=?e Pi5CJ[P87 hf=HM\9Kx

|M Juh\J[9d;>PTJ9UJKV<DT@H;UV 7A4TX JKVU6gJKe[= 7A@H@Hn ;UPTJ;>=HIJ XTJ[6CJ[4TXTJ[4TeKJ MABWV<M\IJMABm;>PFJVUMA@?DF;U=?M\4TVe 7ADTV<JKX XT= e[DT@H;U=?J[Vo=?4 hf=HVUD87:@?=6U[=?4TZ ;>PFJ9UJ[VUDT@ ;>=?4FZ`687:;<;>JK9<4TVs=H4 ;>PT9<JKJpXT=HIJ[4TVU=HM\4TV[x 2mPSDTV;>PTJb;>9UJK7:;>I1JK4L; MAB);UPTJ _r9>7 n(Bacfe[MA;U; IMiXTJ[@8=?V RSDT= ;>Jb@?=HZ\PL; PFJK9UJ!&5TDf;m7]VaniV<;>J[I7 ;>=?e|V<;UDTXFnMAB;>PTJm6T9<M\6gJK9<;U=?J[V MAB ;>PTJ _r9>7 n(BacieKMA;<;dIMiXTJ[@F=?4];>PT9<JKJmXT=HIJ[4TVU=HM\4TV =HV DT4TXTJ[9 7 n\x |M J[hAJK9i=H;e 7A4 5gJ V>7:=?X;UP87:; ;UPTJw;UPT9UJ[JB5XF=?I1JK4TV<=?M\487:@_r9>7 n(BacieKMA;<;mIMiXTJ[@gXF=?VU6F@ 7 n\J[X 5CMA;UPpXT=?V<M\9UXFJK9UJ[X@ 7AI1JK@H@ 7A9CB5@?=HtAJb7A4TX XT9<M\6T@HJ[; B @?=HtAJ 6TPT7AVUJ[VKx

354`;>PTJsV<=?IsDF@ 7:;U=?M\4TVW6T9<JKV<JK4L;>J[X PFJK9UJ JsP87 hAJ DTVUJ[XDu = 0.125

7A4TXDv = 0.05

BEM\9|;UPTJ; MXF=1XCDTVU=HM\4e[M\4TVa; 7A4L;>V 7A4TXè PTMAVUJo;UPTJo687:9>7AI1J[;UJK9UVWBE9UMAI;>PTJ]687A9<;wMAB ;UPTJo6TPT7AVUJ]VU687Ae[J;>P87:;1eK7A4 6F9UMiXTDTeKJV<;U9U=H6CJ[VKx2354 ;>PT=HV V<;>DFXFn JBEMfe[DTVUJ[X2M\4 ;>PFJV<6CJ[eK= 7:@me 7AV<J=H4 WPT=He PVUM\DT9<eKJ[V|MAB e PTJ[I=He 7A@c JK9<J]XF=?V<;U9U=15TDF;>J[X 9>7A4TXFM\I@ n=H4;>PTJYVanfVa;>J[Ix 2mPTJKV<J]V<M\DT9<eKJKVmBEJ[JKX;>PTJWe PTJ[I=He 7A@F;UMr;>PTJWV<niVa;>JKI W= ;>P17reKM\4TVa; 7A4L; 9>7:;UJ%=

+0.01? x\2mPFJ|V<P87A6gJbM:BC;UPTJKV<J|V<M\DT9Ue[JKV

=?V =?4T= ;>=?7A@?@ n|e[9UM\V<V B @?=?t:J< P87 hi=?4TZ|VU=Y@ 5T9>7A4Fe PTJKV =H4Q;UPT9UJ[J XF=?I1JK4TV<=?M\4TV;UM @2:nQ7A4FX UB XT=?9<JKeu;>=?MA4TV9UJKV<6CJ[e[;U=Hh\J[@HnAx ? MA9weK@?7A9U= ;5ngVUJ[J ?_=?Z\DF9UJ 7fx z WPFJK9UJ];>PTJsZA9UMm;>P MAB XFJK4TXT9<=H;UJKV =HV VU=HIsDT@ 7 ;>JKX=?4"; MpXT=?I1JK4FVU=?MA4TVKx 4bMA;>=HeKJ];UP87:; ;UPTJsXTJ[4TXT9<=H;>J[V=H4TXTJKJ[X)V<JKJKI ;>Me[M\4T4TJ[e[;r;UPTJ VUM\DF9UeKJ[VMABcox

kwV I1JK4L;>=HM\4TJKX 5gJ[BEMA9UJ<d;UPTJpIM:;>=Hh:7:;U=?M\4 BEM\9s=H4TeK@HDTXT=?4FZ;>PFJpVUM\DT9<eKJ[V1=HVo;>M=H4ShAJKVa;>=?Z\7:;>J= B;>PTJ 2 DF9U=?4FZI1JKe PT7A4T=?VJ BEM\9YXTJ[VUeK9<=65T=H4TZ4TJKDT9<M\487A@Z\9UMm;>P xQcfPTM\DT@HX";>PT7:;%5gJo;>PFJse 7AV<J< ;UPTJsV<M\DT9Ue[JKVreK7A4 5CJ];>PTM\DFZ\PL;r9UJ[6T9UJ[VUJK4L;U=?4TZp4TJ[DT9UMA4TVWPT=?e P ;>PTJ Z\9<MW=?4TZXTJ[4TXT9U= ;>J[VIsDFV<;1eKM\4F4TJKeu; x 354 ;UPTJV<JK4TV<JM:B 687:;<;>J[9U4 BEM\9UI7:;>=HM\4 =H4AvVanfVa;>J[IVFM\4TJb9UJ[RSDT=?9<JKI1JK4L;l=?V ;>PTJWBEM\9UI7:;U=?M\4 MAB V<;>75T@?Jm;UDM5TDT@?7A9 687:;U;UJK9<4TVKx 2mPTJ%_Y9U7 nBcfeKMA;<;dI1MfXTJ[@Fe[@?J 7:9U@HnQ6T9<MfXFDTeKJ[V ;>PTJ[I&=?4 5CMA;UPs; MQ7A4FXs;>PT9<JKJmXT=HIJ[4TVU=HM\4TV:VUJ[J#?_=?Z\DF9UJf7ix Fx2mPTJpeKM\4F4TJKeu;>JKXF4TJKV<VMABm;UPTJ;>PF9UJKJ B5XT=HIJ[4TVU=HM\487A@ 4TJu; MA9Ut eK7A4T4TMA; 5CJe PFJKe tAJ[X5Ln hi=HVUD87A@7A487A@ nfV<=?V[x bM J[hAJK9587AVUJ[X M\4";UPTJoJ @f;UJK4TV<=HhAJs; M)B XT=?I1JK4FVU=?MA487A@ VU=HIsDT@ 7 ;>=?MA4TV J 5gJK@?=HJ[hAJ;>P87:;m;UPTJYVUM\DF9UeKJ[Vw7:9UJYeKM\4F4TJKeu;>JKX 5LnpXTJK4FXT9U= ;>JKVb7A@HVUM =?4p;UPT9UJ[JQXF=?I1JK4TV<=?M\4TV[x

2mPTJb7A6T6gJ 7A9U7A4TeKJWMABg;>PFJN;>DM5TDF@ 7A9dV<P87A6gJKV VUJ[JKI1Vle PT7A9>7Aeu;>J[9U=?Va;>=HeNBEM\9 ;>PTJ _r9>7 n(Bacfe[MA;U; I1MiXTJK@WPTJK9<J 7AVd= ; 7AV XT= 1eKDT@ ; ;UM MK5F;>7A=?4]=?4o;UPTJ eK7AVUJmMABF;>PTJNZAJK4TJ[9>[email protected] DT9<=?4TZ|V<niV<;UJKIx ?_=HZ\DT9UJO7fx 3VUPTMWV ;>PTJWV<;U9UDTeu;>DT9<JmB 7Ae[;UM\9dMABg;>PTJm;>PF9UJKJ B5XT=HIJ[4TVU=HM\487A@fV<;U9UDTeu;>DT9<J W= ;>P BEDT@H@Hn]Z\9UMW41XTJK4TXVB9U=H;UJKV[x 4TJVUPTMADT@?X 4TMA;U=?eKJ;UP87:;

S(k)=HV RiDF=H;>J V<=?I1=?@ 7:9];>M;>PFJMA4TJeK7A@?e[DT@ 7 ;>JKX BEM\9 ;UPTJ

@ 7AI1JK@H@ 7A9dVa;>9UDFe[;>DF9UJr=?4VUJKeu;>=HM\4 3Tx 3 = ?_=HZ\DT9<J 3Tx9S\7 ?>x |M J[h\J[98;>PFJr7AI16T@?= ;>DTXTJb=?V M\4TJ MA9UXTJ[9MAB_I7AZ\4F=H;>DFXTJ VUI7A@H@?JK9 =?4;>PFJQeK7AVUJQMAB[_Y9U7 nB cfeKM:;U;WI1MfXTJ[@ x

7K7

?_=?Z\DF9UJ 7ix zfj 2mPTJ Z\9UMm;UP MABlXTJK4TXF9U=H;UJKVr=H47 _Y9U7 n(B cfeKMA;<;rI1MfXTJ[@ x 2mPTJ 687:9>7AI1J[;UJK9UV JK9<Je PTM\VUJ[4 ;UM 7Ae[;U=Hh:7:;UJo;>PTJ 6FP87AVUJ 6T9<MfXFDTeK=H4TZpV<;U9U=?6gJKV[x 2mPTJ[9UJ17A9<JsJK=HZ\PL;rVUM\DF9UeKJ[VYMABE5gMA;UP 7A4TX c =?4p;>PTJYVanfVa;>J[Ix

78

?_=?Z\DF9UJ7fx Fj 7 ? 7:;<;>JK9<4"MK5f; 7A=H4TJKX =?47120 × 120

@ 7:;<;>=HeKJ W=H;UP6gJK9<=?MiXT=?e 5gM\DT4FX87A9<neKM\4.BXT=H;U=?M\4TVdDFVU=?4FZ];>PFJ|; M)B5XT=HIJ[4TVU=HM\487A@_Y9U7 n(B cfeKMA;<; I1MiXTJK@8=H41;UPTJw6T9<JKV<JK4Te[J MAB)J[=?Z\PL; VUM\DF9UeKJ[VMAB I1M\9<6TPTM\Z\J[4 c W=H;>P687:9>7AI1J[;UJK9UV

F = 0.065K = .0625

Du = 0.125

Dv = 0.05

x2mPTJ`VUMADT9Ue[JKV 7A6T6gJ 7:9p7AVe[9UM\V<V B @?=?t:J"687 ;U;>J[9U4TV[x 52?4_7:;U;UJK9U4 M5F; 7A=H4TJKX2DTVU=H4TZ ;>PFJ";>PT9<JKJ BXT=?I1JK4TV<=?M\4T7A@ _r9>7 n(Bacfe[MA;U;I1MfXFJK@ W=H;UP BEM\DT9 VUM\DF9UeKJ[V"MABZc>

F = 0.045

K = 0.065

Du = 0.125Dv = 0.05

x

0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

k

|S(k

)|²

?_=?Z\DF9UJ 7ix93Tj 2mPTJpVa;>9<DTe[;UDT9UJB 7Aeu;>M\9]eKM\9<9UJ[VU6gM\4TXT=H4TZ`;>M ? =?Z\DT9<J 7fx 5)x k h\J[9>7AZAJKX M h\JK9 AV>7AI16T@?J[VKx

7K;

9> > = KBD M@D K M RDKOS K@D F

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