PARALLEL LINE ROGUE WAVES OF A (2 + 1)-DIMENSIONAL NONLINEAR SCHR ¨ ODINGER EQUATION DESCRIBING THE HEISENBERG FERROMAGNETIC SPIN CHAIN WEI LIU * College of Mathematics and Information Science, Shandong Technology and Business University, Yantai, 264005, P. R. China E-mail * : [email protected] (corresponding author) Received April 15, 2017 Abstract. Under investigation in this paper is a (2 + 1)-dimensional nonli- near Schr¨ odinger equation describing the Heisenberg ferromagnetic spin chain. A general explicit form of rogue wave solutions for the (2 + 1)-dimensional nonlinear Schr¨ odinger equation is given in terms of the Gram determinants by employing the bilinear method. The relevant formulas involve determinants whose matrix elements are simple polynomials. The fundamental rogue wave is called line rogue wave in the (x,y)-plane, which arises from a constant background with a line profile and then disap- pears into the same background. High-order rogue waves consist of several parallel line rogue waves, and describe the interaction of several fundamental rogue waves. Besides, their dynamical behaviors in the (x,t)-plane are also investigated by three-dimensional plots. Key words: Heisenberg ferromagnetic spin chain, (2 + 1)-dimensional nonlinear Schr ¨ odinger equation, Parallel line rogue waves, Bilinear transformation method. PACS: 02.30.Ik, 02.30.Jr, 05.45.Yv. 1. INTRODUCTION Nonlinear evolution equations (NLEEs) are well used to describe various kinds of nonlinear phenomena in fields such as fluids, plasmas, optics, particle physics, biophysics, and condensed matter physics [1–11]. Exact solutions of NLEEs have been explored to investigate these nonlinear phenomena, because they can give more insight into the physical aspects and then lead to further applications. Indeed, various effective methods have been developed to obtain exact solutions of NLEEs, such as the Darboux transformation method [12, 13], the inverse scattering method [14], the Hirota bilinear method [15], the homogeneous balance method [16, 17], the Lie group method [18, 19], and so on [20–22]. Originally, the rogue waves were observed in a relatively calm sea, where they suddenly appear from nowhere as localized and isolated surface waves, and then make a sudden hole in the sea just before attaining surprisingly high amplitudes, and Romanian Journal of Physics 62, 118 (2017) v.2.0*2017.9.13#9586ba22
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PARALLEL LINE ROGUE WAVES OF A (2+1)-DIMENSIONAL NONLINEARSCHRODINGER EQUATION DESCRIBING THE HEISENBERG
FERROMAGNETIC SPIN CHAIN
WEI LIU*
College of Mathematics and Information Science, Shandong Technology and Business University,Yantai, 264005, P. R. China
Abstract. Under investigation in this paper is a (2 + 1)-dimensional nonli-near Schrodinger equation describing the Heisenberg ferromagnetic spin chain. Ageneral explicit form of rogue wave solutions for the (2+ 1)-dimensional nonlinearSchrodinger equation is given in terms of the Gram determinants by employing thebilinear method. The relevant formulas involve determinants whose matrix elementsare simple polynomials. The fundamental rogue wave is called line rogue wave in the(x,y)-plane, which arises from a constant background with a line profile and then disap-pears into the same background. High-order rogue waves consist of several parallel linerogue waves, and describe the interaction of several fundamental rogue waves. Besides,their dynamical behaviors in the (x,t)-plane are also investigated by three-dimensionalplots.
Nonlinear evolution equations (NLEEs) are well used to describe various kindsof nonlinear phenomena in fields such as fluids, plasmas, optics, particle physics,biophysics, and condensed matter physics [1–11]. Exact solutions of NLEEs havebeen explored to investigate these nonlinear phenomena, because they can give moreinsight into the physical aspects and then lead to further applications. Indeed, variouseffective methods have been developed to obtain exact solutions of NLEEs, suchas the Darboux transformation method [12, 13], the inverse scattering method [14],the Hirota bilinear method [15], the homogeneous balance method [16, 17], the Liegroup method [18, 19], and so on [20–22].
Originally, the rogue waves were observed in a relatively calm sea, where theysuddenly appear from nowhere as localized and isolated surface waves, and thenmake a sudden hole in the sea just before attaining surprisingly high amplitudes, and
Romanian Journal of Physics 62, 118 (2017) v.2.0*2017.9.13#9586ba22
Article no. 118 Wei Liu 2
finally disappear without a trace. Recently, efforts devoted to study such rare ex-treme events have ranged from oceanography [23] to various areas, such as nonlinearoptics [24–26], Bose-Einstein condensates [27], plasma physics [28], and so on [29–32]. Mathematically, rogue waves are expressed in rational form, and are localizedin both space and time [33, 34]. Recently, a variety of nonlinear soliton equationsincluding nonlocal systems satisfying parity-time (PT ) symmetry have been verifiedpossessing rogue wave solutions [35–54]. Two recent articles [55, 56] have providedcomprehensive overviews on various rogue-wave phenomena from the physical pointof view.
Compared with one-dimensional nonlinear systems, the higher dimensionalnonlinear systems are less studied mainly because of the non-availability of analyti-cal methods like in one dimension. However, from the realistic and physical pointof views, the extension of rogue waves studies to higher dimensions is essential be-cause the physical fields are modeled by multi-dimensional systems, such as in thestudies of ocean waves and ultrafast nonlinear optics. In particular, ultrafast opticalrogue waves are also higher dimensional, because the spatial and temporal degreesof freedom cannot be treated separately in the theoretical and experimental studiesof self-focusing of intense ultrashort pulses [57–64]. Up to now, different types ofrogue waves occurring in a series of multi-dimensional systems have been investi-gated [65–72].
Nonlinear magnetization dynamics of the Heisenberg ferromagnetic spin chainwith different magnetic interactions in classical and semiclassical limit has been as-sociated with soliton theory and condensed matter physics [73–75]. Nonlinear spinexcitations in the magnetic materials have their applications in the microwave com-munication systems and nonlinear signal processing devices [76, 77]. The dynamicsof the nonlinear spin excitations in the Heisenberg ferromagnetism can be describedby the nonlinear Schrodinger-type equations [78]. Theoretically, the nonlinear dy-namics of the (2+1)-dimensional ferromagnetic spin systems with bilinear and bi-quadratic interactions in the semiclassical limit has been discussed, which has beendescribed by integrable (2+1)-dimensional nonlinear Schrodinger (NLS) equations[79–84]. In this paper, we will consider a (2+1)-dimensional NLS equation
iut− iux+uxx+uyy−2uxy+2|u|2u= 0, (1)
for the (2+ 1)-dimensional Heisenberg ferromagnetic spin chain with bilinear andanisotropic interactions in the semiclassical limit. Here u(x,y, t) is a complex-valuedfunction, x,y, and t, respectively, denote the scaled spatial and time coordinates.
For the (2 + 1)-dimensional NLS equation (1), the Lax pair has been con-structed, and the bright soliton solutions have been obtained by virtue of the Hirotamethod [78] and Darboux transformation [82], and dark soliton interactions have alsobeen discussed [84]. To the best of our knowledge, general high-order rogue waves
3 Parallel rogue waves of (2+1)-D NLSE for Heisenberg ferromagnetic spin chain Article no. 118
have not been investigated for the (2+1)-dimensional NLS equation (1).In this work, we obtain parallel line rogue waves for the (2+1)-dimensional
NLS equation (1), which are expressed in term of determinants by employing theHirota’s bilinear method [15]. The basic idea is to treat the (2+1)-dimensional NLSequation (1) as a constrained KP hierarchy, then rational solutions of the equation (1)are reduced from tau functions of the KP hierarchy [18, 85].
The outline of the paper is organized as follows. In Sec. 2, the exact andexplicit rational solutions of the (2+1)-dimensional NLS equation (1) are presentedin the determinant form by using the Hirota bilinear method. In Sec. 3, the dynamicsof rogue wave solutions is analyzed and illustrated graphically. Section 4 contains asummary and discussion.
2. ROGUE WAVE SOLUTIONS VIA DETERMINANTS OF N ×N MATRICES
In this Section, we derive an explicit form of the rogue wave solutions for the(2+1)-dimensional NLS equation (1). Using the dependent variable transformation
u= e2itg
f, (2)
the (2+1)-dimensional NLS equation can be transformed into the bilinear forms
(iDt+D2x+D
2y−2DxDy)g ·f = 0,
(D2x+D
2y−2DxDy)f ·f = 2(gg∗−f2).
(3)
Here, f is a real function, g is a complex function, the asterisk denotes complexconjugation, and the operator D is the Hirota’s bilinear differential operator [15]defined by
where P is a polynomial of Dx, Dy, Dt, · · ·.Theorem 1. The (2+1)-dimensional NLS equation (1) has rogue wave solutions (2)with f and g given by N ×N determinants
f = τ0 , g = τ1, (4)
where τn = det1≤i,j≤N (m(n)2i−1,2j−1), and the matrix elements are given by
p y+ px+2(ip2 + p)t, i, j are arbitrary positive integers, ak and al arearbitrary complex constants.
By a scaling of mij , we can normalize a0 = 1 without loss of generality, thushereafter we set a0 = 1 in this paper. Note that these rational solutions can also beexpressed in terms of Schur polynomials as discussed in [37, 65]. Next we give ashort proof of Theorem 1.Lemma 1. The bilinear equation in the KP hierarchy
(D2x1−Dx2)τn+1 · τn = 0,
(Dx−1Dx1−2)τn · τn =−2τn+1 · τn,(6)
has the Gram determinant solutions
τn = det1≤i,j≤N
(m(n)ij ), (7)
with the matrix element m(n)ij satisfying the following differential and difference re-
lations,
∂x1m(n)ij = ψ
(n)i φ
(n)j ,
∂x2m(n)ij = ψ
(n+1)i φ
(n)j +ψ
(n)i φ
(n−1)j ,
∂x−1m(n)ij =−ψ(n−1)
i φ(n+1)j ,
m(n+1)ij =m
(n+1)ij +ψ
(n)i φ
(n+1)j ,
∂xvψi = ψ(n+v)i ,
∂xvφj = φ(n−v)j (v =−1,1,2).
(8)
This Lemma can been proved in a similar way as the proof of the Lemma 1 ofRefs. [36, 85], thus we omit the proof of this Lemma in this paper. Next we use thisLemma to prove Theorem 1.Proof of Theorem 1. To get rational solutions for the (2 + 1)-dimensional NLSequation, we choose functions m(n)
5 Parallel rogue waves of (2+1)-D NLSE for Heisenberg ferromagnetic spin chain Article no. 118
where
Ai =
i∑k=0
ak(i−k)!
(p∂p)i−k, Bj =
j∑l=0
bl(j− l)!
(q∂q)j−l,
ξ =1
px−1+px1+p
2x2 , η =1
qx−1+ qx1− q2x2 ,
and for simplicity, the functions m(n)ij can be rewritten as
m(n)i,j =eξ+η(−
p
q)n
ni∑k=0
ak
(i−k)!(p∂p+ ξ
′+n)ni−k
nj∑l=0
bl
(j− l)!(q∂q+η
′−n)nj−l 1
p+ q, (10)
where
ξ′=−1
px−1+px1+2p2x2 , η
′=−1
qx−1+ qx1−2q2x2.
Here p, q, ak , bl are arbitrary complex constants, and i, j,ni, N are arbitrary positiveintegers.
Further, taking the parameter constraints
q = p∗ = 1, bk = a∗k, (11)
and assuming x1 ,x3 are real, x2 is pure imaginary, we have
η′= ξ
′∗, m∗ij(n) =mji(−n), τ∗n = τ−n. (12)
What is more, under parameter constraints (11), the τn satisfies the reduction condi-tion
(∂x1 +∂x−1)τn = 4Nτn (13)
Applying the change of independent variables
x−1 =−y,x1 = 2x+2t ,x2 = i t,
and taking
τ(0) = f,τ(1) = g,τ(−1) = g∗,
the bilinear equation (6) can be transformed into the bilinear equation (3). Finally,using the gauge freedom of τn, it is easy to get the rational solutions of the (2+1)-dimensional NLS equation (1) from the solutions of equation (6), which are given inTheorem 1. Thus the Theorem 1 has been proved.
Below we concentrate on commenting that the obtained solutions are nonsin-gular by using equation (8) , (9), and (11). We note that f = τ0 is given by thedeterminant
7 Parallel rogue waves of (2+1)-D NLSE for Heisenberg ferromagnetic spin chain Article no. 118
where p = 1, and a1 is a freely complex constant. After a shift of time and spacecoordinates, a1 can be eliminated and the fundamental rogue wave can be written as
u= e2it[1− 16it+4
(4x+2y+4t−1)2+16t2+1]. (17)
It follows that |u| has three extreme lines given by
L1 : y =−2x−2t+1
2,
L2 : y =−2x−2t+1
2+
1
2
√48t2+3,
L3 : y =−2x−2t+1
2− 1
2
√48t2+3.
(18)
Based on the analysis of critical values for rational solution |u| (17), the maximumamplitude of |u| is given by
|u|max = |u| |L1=
√1+
8
16t2+1
and the minimum amplitude of |u| is given by
|u|min = |u| |L2= |u| |L3=
√1− 1
16t2+1.
We define Lwidth as a distance between L2 and L3, thus
Lwidth =
√48t2
5.
It is not easy to find that |u|max → |u|min = 1 and Lwidth →∞ when |t| → ∞,which means that the rational solution u defined in (17) approaches to a constantbackground as |t| goes to∞, and has a large amplitude for only a short period of time.Thus solution u defined in (17) is a line rogue wave. This solution is shown in Fig.1. As can be seen, this solution possesses a line profile with a varying height, whichis different from the moving line solitons of the multi-dimensional soliton equations.Line solitons maintain a perfect profile without any decay during their propagationin the (x,y)-plane, but the solution |u| approaches the constant background when|t|>> 0, whereas at intermediate times it reaches a much higher amplitude. Note that|u| reaches the maximum amplitude 3 (i.e., three times the background amplitude)along the line L1, and the minimum amplitude 0 along lines L2 and L3 at t= 0.
The above discussion focused on the fundamental rogue wave solutions of the(2+1)-dimensional NLS equation. Below we will discuss the high-order rogue wavesolutions corresponding to those given in Theorem 1, for N ≥ 2.
Fig. 1 – (Color online) A fundamental line rogue wave |u| given by (17), for the (2+1)-dimensionalNLS equation (1), plotted in the (x,y)-plane. Here the parameter a1 = 0.
3.2. HIGH-ORDER ROGUE WAVE SOLUTIONS
For an arbitrary given value of N , the N th-order rogue waves for the (2+1)-dimensional NLS equation (1) can be generated by using the results of Theorem 1.These rogue wave solutions have different dynamics in the (x,y)-plane and in the(x,t)-plane or (y,t)-plane. In the (x,y)-plane, the N th-order rogue waves consist ofN individual fundamental line rogue waves, and these line rogue waves are parallellines. That is different from nonfundamental rogue waves in the Davey-Stewartson(DS) equations [65, 66] and Fokas systems [70], since the latter are no longer linewaves. In the (x,t)-plane or (y,t)-plane, the N th-order rogue waves are made up ofN(N+1)
2 localized waves, and these localized waves feature as (1+1)-dimensionalrogue waves.
To demonstrate high-order rogue waves, we first consider the case of N = 2.
−192 itx−96 iyt−6144 it5+3072 it4−384 it3−192 it2+120 it.
This solution is shown in Fig. 2. It is seen that when two parallel line roguewaves arise from the constant background in the (x,y)-plane, the region of theirintersection acquires higher amplitude first (see the panel at t = −2). At the inter-mediate time, the superposition of two parallel line rogue waves generates one main
Fig. 2 – (Color online) The second-order rogue wave |u| given by (21), of the (2+1)-dimensional NLSequation (1), plotted in the (x,y)-plane.
peak posing maximum amplitude and several lower peaks (see the panel at t= 0). Atlarger time, the two parallel line rogue waves decay back to the constant background(see the panel at t = 5). It is noticed that the maximum of |u| is equal to 5 (i.e.,five times the height of the background), which is much higher than the second-orderrogue waves in the DS systems [65, 66], since the latter does not exceed four timesthe constant background for all the time.
11 Parallel rogue waves of (2+1)-D NLSE for Heisenberg ferromagnetic spin chain Article no. 118
(a) (b)
Fig. 3 – (Color online) The second-order rogue wave |u| given by (21), of the (2+1)-dimensional NLSequation (1), with the parameters (a) y = 0, and (b) y = 10, plotted in the (x,t)-plane.
Typical features of the second-order rogue wave solution in the (x,t) plane areshown in Fig. 3. Visually, this solution consists of three fundamental rogue wavesas second-order rogue waves in (1 + 1)-dimensional systems. The interaction ofthese three rogue waves can also generate some common wave patterns, such as thefundamental pattern (see panel (a) of Fig. 3), and the triangular pattern (see panel (b)of Fig. 3).
Next, we proceed to consider more complicated cases of Theorem 1. For in-stance, with N = 3, and parameter choices
a0 = 1,a1 = 0,a2 = 0,a3 =−1
12,a4 = 0,a5 =−
1
240, (22)
the corresponding solutions are shown in Figs. 4 and 5. Comparing to the second-order rogue wave shown in Figs. 2 and 3, wave patterns of the third-order rogue waveare more complicated. In the (x,y)-plane, the third-order rogue wave consist of threeindividual fundamental line rogue wave, which arise from the constant backgroundand then decay back to the constant background at larger time, see Fig. 4. It isnoticed that in this whole process, these three fundamental line wave keep parallel,and the maximum value of solution |u| is equal to 7 (i.e., seven times the heightof the background). Thus the interaction between these three line rogue waves cangenerate very high peaks. In the (x,t)-plane, the third-order rogue waves for the(2+1)-dimensional NLS equation are composed by 6 fundamental localized wavesas third-order rogue waves in (1+1)-dimensional soliton equations, see Fig. 5. Threebasic patterns, namely the fundamental pattern, the triangular pattern, and the circular(ring) pattern, are also exhibited (see panels (a), (b), and (c) of Fig. 5, respectively).
For larger N , these high-order waves have qualitatively similar behaviors, be-sides that more line rogue waves will arise and interact with each other, and thetransitional profiles of these solutions become much more complicated. Motivated
Fig. 4 – (Color online) The third-order rogue wave |u| of the (2+1)-dimensional NLS equation (1),with the parameters N = 3,a1 = 0,a2 = 0,a3 =− 1
12 ,a4 = 0,a5 =− 1240 , plotted in the (x,y)-plane.
by the key features of rogue waves up to third-order, which are shown in Figs. 1-5,higher-order rogue waves |u| given in Theorem support the following conjectures:(1) In the (x,y)-plane, the N th-order parallel line rogue waves consist of N indi-vidual fundamental line rogue waves. The maximum values of |u| is 2N +1 (i.e.,2N +1 times the height of the background). For the fundamental pattern, the N th-order rogue wave has n(n+1)− 1 peaks, and the central peak is surrounded by
13 Parallel rogue waves of (2+1)-D NLSE for Heisenberg ferromagnetic spin chain Article no. 118
(a) (b)
(c)
Fig. 5 – (Color online) Three patterns of the third-order rogue wave solution |u| of the (2 + 1)-dimensional NLS equation (1), plotted in the (x,t)-plane. (a) The fundamental pattern with parametersN = 3,a1 = 0,a2 = 0,a3 = − 1
12 ,a4 = 0,a5 = − 1240 ,y = 0. (b) The triangular pattern with para-
meters N = 3,a1 = 0,a2 = 0,a3 = 20,a4 = 0,a5 = 0,y = 0. (c) The ring pattern with parametersN = 3,a1 = 0,a2 = 0,a3 = 0,a4 = 0,a5 = 10i,y = 0.
N(N +1)−2 gradually decreasing peaks.(2) In the (x,t)-plane or (y,t)-plane, the N th-order rogue waves are made up ofN(N+1)
2 rogue waves as in (1+1)-dimensional soliton equations. They also possesssome common patterns, namely fundamental pattern, triangular, and circular pat-terns. For the triangular and circular patterns, there are N(N +1)/2 uniform peaks.For the circular pattern, an order-N rogue wave displays a ring structure, the outerring has 2N − 1 uniform peaks, and the inner structure is an order-(N − 2) roguewave.
4. SUMMARY AND DISCUSSION
In summary, we have derived a general formula for the N -th order rogue wavesolutions of the (2 + 1)-dimensional NLS equation (1) by employing the bilineartransformation method in Theorem 1, in which solutions are expressed explicitly
in terms of determinants. The first-order rogue wave solution is given explicitly inEq. (17). The asymptotic behaviors of this solution are analyzed, and the typicalevolution dynamics is shown in Fig. 1. It is seen that the fundamental rogue wave isa line rogue wave that arises from the constant background and then disappears intothe constant background. The explicit form of the second-order rogue wave solutionis given in Eq. (21). Figures 2 and 3 display the dynamical features of the second-order rogue waves, which consist of three parallel line rogue waves. The third-orderrogue waves are composed of three parallel line rogue waves in the (x,y)-plane, andthese line waves just exist on the constant background for a short period of time, seeFig. 4. The dynamics of the third-order rogue waves in the (x,t)-plane is shown inFig. 5. The main features of higher-order rogue waves are summarized as follows:(1) In the (x,y)-plane, the N th-order rogue waves consist of N parallel line roguewaves.(2) In the (x,t)-plane or the (y,t)-plane, the N th-order rogue waves are composedof N(N+1)
2 localized waves as in (1+1)-dimensional soliton equations.
REFERENCES
1. C. Hamner, J. J. Chang, P. Engels, and M. A. Hoefer, Phys. Rev. Lett. 106, 560 (2010).2. H. J. Jiang, J. J. Xiang, C. Q. Dai, and Y. Y. Wang, Nonlinear Dyn. 75, 201 (2014).3. L. Wang, J. H. Zhang, C. Liu, M. Li, and F. H. Qi, Phys. Rev. E 93, 062217 (2016);
L. Wang, J. H. Zhang, Z. Q. Wang, C. Liu, M. Li, F. H. Qi, and R. Guo, Phys. Rev. E 93, 012214(2016);L. Wang, Y. J. Zhu, Z. Q.Wang, T. Xu, F. H. Qi, and Y. S. Xu, J. Phys. Soc. Jpn. 85, 024001 (2016);L. Wang, Y. J. Zhu, F. H. Qi, M. Li, and R. Guo, Chaos 25, 063111 (2015);L. Wang, C. Geng, L. L. Zhang, and Y. C. Zhao, EPL 108, 50009 (2014);L. Wang, X. Li, F. H. Qi, and L. L. Zhang, Ann. Phys. 359, 97 (2015);L. Wang, M. Li, F. H. Qi, and T. Xu, Phys. Plasmas 22, 032308 (2015).
4. D. W. Zuo, Y. T. Gao, Y. J. Feng, and L. Xue, Nonlinear Dyn. 78, 2309 (2014).5. Z. Z. Lan, Y. T. Gao, J. W. Yang, C. Q. Su, and B. Q. Mao, Commun. Nonlinear Sci. Numer.
Simulat. 44, 360 (2017).6. M. M. Rajan and A. Mahalingam, Nonlinear Dyn. 79, 2469 (2015).7. D. Mihalache, Rom. J. Phys. 59, 295 (2014).8. V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, Rom.
Rep. Phys. 67, 5 (2015).9. D. S. Wang, Y. S. Xue, and Z. F. Zhang, Rom. J. Phys. 61, 827 (2016).
10. B. Malomed, L. Torner, F. Wise, and D. Mihalache, J. Phys. B: At. Mol. Opt. Phys. 49, 170502(2016).
11. P. G. Kevrekidis and D. J. Frantzeskakis, Reviews in Physics 1, 140 (2016).12. V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer, Berlin, 1991.13. C. H. Gu, H. S. Hu, and Z. X. Zhou, Darboux Transformation in Soliton Theory and Geometric
Applications, Shanghai Science and Technology Press, Shanghai, 1999.
15 Parallel rogue waves of (2+1)-D NLSE for Heisenberg ferromagnetic spin chain Article no. 118
14. M. J. Ablowitz and P. A. Clarkon, Solitons, Nonlinear Evolution and Inverse Scattering, Cam-bridge University Press, Cambridge, UK, 1991.
15. R. Hirota, The direct method in soliton theory, Cambridge University Press, Cambridge, UK, 2004.16. V. A. Vladimirov and C. Maczka, Rep. Math. Phys. 60, 317 (2007).17. M. T. Darvishi, M. Najafi, L. Kavitha, and M. Venkatesh, Commun. Theor. Phys. 58, 785 (2012).18. M. Jimbo and T. Miwa, Publ. Res. Inst. Math. Sci. 19, 943 (1983).19. Z. H. Zhao and Z. D. Dai, Int. J. Nonlinear Sci. Numer. Simul. 11, 677 (2010).20. A. M. Wazwaz, Appl. Math. Comput. 317, 72 (2017).21. H. Triki and A. M. Wazwaz, Nonlinear Sci. Numer. Simul. 9, 404 (2014).22. A. M. Wazwaz, R. Rach, and J. S. Duan, Appl. Math. Comput. 29, 5004 (2013).23. C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue waves in the Ocean, Springer, 2009.24. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).25. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, F. Genty, N. Akhmediev, and J. M. Dudley, Nat.
Phys. 6, 790 (2010).26. J. S. He et al., Rom. J. Phys. 62, 203 (2017).27. Y. V. Bludov, V. V. Konotop, and N. Akhmediev, Phys. Rev. A 80, 033610 (2009).28. H. Bailung, S. K. Sharma, and Y. Nakamura, Phys. Rev. Lett. 107, 255005 (2011).29. M. Shats, H. Punzmann, and H. Xia, Phys. Rev. Lett. 104, 104503 (2010).30. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, Phys. Rep. 528, 47 (2013).31. A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Phys. Rev. Lett. 106, 204502 (2011).32. A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. X 2, 011015 (2012).33. D. H. Peregrine, J. Aust. Math. Soc. B 25, 16 (1983).34. N. Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A 373, 675 (2009).35. Z. X. Xu and K. W. Chow, Appl. Math. Lett. 56, 72(2016).36. Y. Ohta and J. Yang, Proc. R. Soc. London, Ser. A 468, 1716 (2012).37. Y. S. Tao and J. S. He, Phys. Rev. E 85, 026601 (2012).38. S. H. Chen, Phys. Rev. E 88, 023202 (2013).39. G. Mu, Z. Y. Qin, and R. Grimshaw, SIAM J. Appl. Math. 75, 1 (2015).40. J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian, and A. S. Fokas, Phys. Rev. E 87, 052914 (2013).41. L. H. Wang, K. Porsezian, and J. S. He, Phys. Rev. E 87, 053202 (2013).42. W. Liu, D. Q. Qiu, and J. S. He, Commun. Theor. Phys. 63, 525 (2015).43. Jing Yu, Jingwei Han, and Jingsong He, Z. Naturforsch. A 70, 1039 (2015).44. Z. Y. Yan, V. V. Konotop, and N. Akhmediev, Phys. Rev. E 82, 036610 (2010).45. Z. Y. Yan, Phys. Lett. A 375, 4274 (2011).46. S. Chen and D. Mihalache, J. Phys. A: Math. Theor. 48, 215202 (2015).47. S. H. Chen, J. M. Soto-Crespo, F. Baronio, P. Grelu, and D. Mihalache, Opt. Express 24, 15251
(2016).48. S. H. Chen, P. Grelu, D. Mihalache, and F. Baronio, Rom. Rep. Phys. 68, 1407 (2016).49. X. Wang, Y. Q. Li, F. Huang, and Y. Chen, Commun. Nonlinear Sci. Numer. Simulat. 20, 434
(2015).50. X. Wang, J. L. Cao, and Y. Chen, Phys. Scr. 90, 105201 (2015).51. G. Mu and Z. Y. Qin, Nonlinear Anal. Real World Appl. 18, 1 (2014).52. N. Akhmediev et al., J. Opt. 18, 063001 (2016).53. S. W. Xu, K. Porsezian, J. S. He, and Y. Cheng, Rom. Rep. Phys. 68, 316 (2016).54. A. Ankiewicz and N. Akhmediev, Rom. Rep. Phys. 69, 104 (2017).
55. D. Mihalache, Rom. Rep. Phys. 67, 1383 (2015).56. D. Mihalache, Rom. Rep. Phys. 69, 403 (2017).57. F. Baronio, S. Wabnitz, and Y. Kodama, Phys. Rev. Lett. 116, 173901 (2016).58. P. H. Pioger, V. Couderc, L. Lefort, A. Barthelemy, F. Baronio, C. De Angelis, Y. Min, V. Quiring,
and W. Sohler, Opt. Lett. 27, 2182 (2002).59. S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, M. Mysyrowicz, A. Couairon, and L. Berge, Phys.
Rev. Lett. 87, 213902 (2001).60. H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, Phys. Rev.
Lett. 87, 043902 (2001).61. J. K. Ranka, R. W. Schirmer, and A. L. Gaeta, Phys. Rev. Lett. 77, 3783 (1996).62. X. Liu, L. J. Qian, and F. W. Wise, Phys. Rev. Lett. 82, 4631 (1999).63. H. Leblond and D. Mihalache, Phys. Rep. 523, 61 (2013).64. D. J. Frantzeskakis, H. Leblond, and D. Mihalache, Rom. J. Phys. 59, 767 (2014).65. Y. Ohta and J. K. Yang, Phys. Rev. E 86, 036604 (2012).66. Y. Ohta and J. K. Yang, J. Phys. A: Math. Theor. 46, 105202 (2013).67. F. Yuan, J. G. Rao, K. Porsezian, D. Mihalache, and J. S. He, Rom. J. Phys. 61, 378 (2016).68. Y. B. Shi and Y. Zhang, Nonlinear Sci. Numer. Simul. 44, 120 (2017).69. P. Dubard and V. B. Matveev, Nonlinearity 26, 93 (2013).70. J. G. Rao, L. H. Wang, Y. Zhang, and J. S. He, Commun. Theor. Phys. 64, 605 (2015).71. C. Qian, J. G. Rao, Y. B. Liu, and J. S. He, Chin. Phys. Lett. 33, 110201 (2016).72. Y. B. Liu, A. S. Fokas, D. Mihalache, and J. S. He, Rom. Rep. Phys. 68, 1425 (2016).73. M. Lakshmanan, Phys. Lett. A 61, 53 (1977).74. L. A. Takhtajan, Phys. Lett. A 64, 235 (1977).75. J. Tjon and J. Wright, Phys. Rev. B 15, 3470 (1977).76. P. E. Wigen, Nonlinear Phenomena and Chaos in Magnetic Materials, World Sci., Singapore,
1994.77. A. N. Slavin and I. V. Rojdestvenski, IEEE Trans. Magn. 30, 37 (1994).78. G. L. Lamb, Phys. Rev. Lett. 37, 235 (1976).79. D. Y. Liu, B. Tian, Y. Jiang, X. Y. Xie, and X. Y. Wu, Comput. Math. Appl. 71, 2001 (2016).80. C. C. Vasanthi and M. M. Latha, Commun. Nonl. Sci. Numer. Simul. 28, 109 (2015).81. M. M. Latha and T. Anitha, Phys. B 479, 149 (2015).82. Q. M. Wang, Y. T. Gao, C. Q. Su, B. Q. Mao, Z. Gao, and J. W. Yang, Ann. Phys. 363, 440 (2015).83. M. M. Latha and C. C. Vasanthi, Phys. Scr. 89, 065204 (2014).84. X. H. Zhao, B. Tian, D. Y. Liu, X. Y. Wu, J. Chai, and Y. J. Guo, Superlattice Microst. 100, 587
(2016).85. Y. Ohta, D. Wang, and J. Yang, Stud. Appl. Math. 127, 345 (2011).
