Wave Collapse in Nonlocal Nonlinear Schrödinger …Nonlinear Physics. Theory and Experiment IV 2006...

Nonlinear Physics. Theory and Experiment IV 2006

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Wave Collapse in NonlocalNonlinear Schrödinger

EquationsĐ. BAKIRTA Ş

ĐTÜ DEPARTMENT OF MATHEMATICS

M. J. ABLOWITZ *, B. ILAN *** CU DEPARTMENT OF APPLIED MATHEMATICS

** UC MERCED DEPARTMENT OF APPLIED MATHEMATICS

Ablowitz et al. Physica D 207 (2005) 230-253


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COLLAPSECOLLAPSE

• The solutions of nonlinear wave equations often exhibit important phenomena such as stable localized waves (e.g. solitons), self similar structures, chaotic dynamics and wave singularities such as shock waves (derivative discontinuities) and/or wave collapse (i.e, blow up) where the solution tends to infinity in finite time or finite propagation distance.

• Nonlinear wave collapse is a matter of interest in many areas ofphysics, hydrodynamics and optics.

• A prototypical equation that arises in cubic media, such as Kerr media in optics, is the (2+1)D focusing cubic nonlinear Schrödinger equation NLS

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0

1( , , ) ( ) 0, ( , ,0) ( , )

2z xx yyiu x y z u u u u u x y u x y+ + + = =


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NonlinearNonlinear SchrSchröödingerdinger EquationEquation & &

CollapseCollapse

• Kelley (1965) carried out direct numerical simulations of cubic NLS that indicated the possibility of wave collapse.

• Vlaslov et al. (1970) proved that the solutions of the cubic NLS satisfy the Virial Theorem (Variance Identity)

2 22 22 ( ) 4 ,

dx y u H

dz+ =∫

2 4

0 0

1( )

2H u u= ∇ −∫Hamiltonian:

They also concluded that the solution of the NLS can become singular in finite time (or distance) because a positive quantity could become negative for initial conditions satisfying .0H <


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Subsequently many researchers have studied the NLS in detail:

• Weinstein (1983) showed that when the power is sufficiently small, i.e.,

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0 1.8623cN u const N π= = < ≈∫The solution exists globally.

Therefore, the sufficient condition for collapse is 0H <While the necessary condition for collapse is cN N>

Weinstein also found that the ground state of the NLS also plays animportant role in the collapse theory. The ground state is a “stationary” solution of the form ( ) izu R r e=


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• Papanicolaou et al. (1994) studied the singularity structure near the collapse point and showed asymptotically and numerically that colapse occurs with a (quasi) self-similar profile.

• Merle and Raphael (1996) elaborated on the behavior of blow up phenomena of NLS.

• Gaeta et al. (2000) carried out detailed experiments which reveal the nature of the singularity formation and showed that collapse occurs with a self-similar profile.


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There are considerably fewer studies of the wave collapse that arise in nonlinear media whose governing equations have quadraticnonlinearities,

such as water waves and nonlinear optics.

The derivation of the NLSM system is based on an expansion of the slowly-varying wave amplitude in the first and second harmonics of the fundamental frequency, as well as a mean term that corresponds to the zeroth harmonic.

This leads to a system of equations that describes the nonlocal-nonlinear coupling between a dynamic field that is associated with the first harmonic and a static field associated with the mean term.

(2)χ


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φ

2( , ) ( , , )G x x y y u x y z dx dy

xφ

∞

−∞

∂′ ′ ′ ′ ′ ′= − −′∂∫

2( )xx yy xuφ νφ+ =

For the physical models considered in this study, the general nonlinear Schrödinger-mean (NLSM) system can be written in the following form

( ) 2

1 2

10

2t xx yy xiu u u u u uσ σ ρ φ+ + + − =

These equations are also sometimes referred to as BenneyBenney--RoskesRoskesor DaveyDavey--StewartsonStewartson type and are nonlocal because the second equation can be solved for

)/log()4(),( 221 νπ yxyxG += −

Which corresponds to a strongly-nonlocal function φ


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• NLSM equations were originally obtained by Benney and Roskes (1969) in their study of the instability of wave packets in multidimensional water wavepackets in water of finite depth, without surface tension.

• Davey and Stewartson (1974) derived a special form of NLSM equations in the study of water waves, near the shallow water limit.

• Djordjevic and Redekopp (1977) extended the results of Benney andRoskes to include the surface tension.

• Ablowitz and Segur (1979) analyzed the Benney- Roskes equations andshowed that the singularity exists in some parameter regimes.They further introduced the Hamiltonian of NLSM system.

• Existence and well-posedness of solutions to NLSM equations was studiedby Ghidaglia and Saut (1990)

NLSM EQUATION FROM WATER WAVESNLSM EQUATION FROM WATER WAVES


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Free-surface gravity-capillary water waves NLSM results from a weaklynonlinear quasi-monchromatic expansion of velocity potential as

( ) 2 2 ( )2( , , ) ~ [ . . ] [ . .] ...i ikx t kx tx y t Ae c c A e c cω ωφ ε ε− −+ + Φ + + +% %%

Substituting the wave expansion into Euler’s equations with a free surfaceand assuming slow modulations of the field in and directions results a nonlinearly coupled system for and .

x yA% Φ%

x : direction of propagation : transverse direction

: time : measure of the weak nonlinearity

yt ε

: coefficients of the zeroth, first, second harmonics2, ,A AΦ % %%

Derivation of NLSM in water waves


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• In the context of water waves,Ablowitz and Segur (1979), studied theNLSM (Benney-Roskes) Equations in the following form

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1iA A A A A Aτ ξξ ηη ξλ µ χ χ+ + = + Φ2( )Aξξ ηη ξα βΦ + Φ = −

2 1/ 2( ), , ( )gk x c t ly gk tξ ε η ε τ ε= − = =where

Dimensionless coord.,

( , )k l are the wave numbers in the directions,

gcgroup velocity1, , , , ,λ µ χ χ α β are suitable functions of :

wave number, dispersion coefficients

hnormalized water depth

2 2/ ,kω∂ ∂ 2 2/ lω∂ ∂and surface tension ,T 1, 0µ χ ≥where

( , )x y


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( ) 2

1 2

10

2t xx yy xiu u u u u uσ σ ρ φ+ + + − =

2( )xx yy xuφ νφ+ =

1 2 1 0σ σ ν= = >For

By rescaling the variables, previous system can be transformed to

(Elliptic-elliptic case), this system admits Collapse, requires large surface tension


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HamiltonianHamiltonian & Virial Theorem& Virial Theorem• Ablowitz&Segur (1979) defined the Hamiltonian

2 24 2 21 11

( ) ( ) ( )2

A AH A d dξ η

αχ χλ µ χ ξ ηξ η β β

∂ ∂ = + − − + Φ + Φ ∂ ∂ ∫∫

2 2 22

2 8A d d Hξ η ξ η

τ λ µ ∂ + = ∂

∫∫

Furthermore, they showed that the VirialVirial TheoremTheorem holds

As can be seen if H <0, the moment of inertia vanishes at a finite time and

no global solution exists after this time. This indicates a rapid development of

singularity by which we mean the

Each bracket, { }, in H is positive definite, and the second bracket vanishes

in the linear limit of Benney Roskes equations. Clearly H<0 is possible.

FOCUSINGFOCUSING.


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NLSM EQUATION FROM OPTICSNLSM EQUATION FROM OPTICS

• In isotropic (Kerr) media, where the nonlinear response of the material depends cubically on the applied field, the dynamics of a quasi-monochromatic optical pulse is governed by the NLS equation.

• Generalized NLS systems with coupling to a mean term also appear in various physical applications. These equations are denoted as NLSM typeequations. NLSM type equations arise in nonlinear optics by studying materials with quadratic nonlinear response.

• Ablowitz, Biondini and Blair (1997, 2001) found that NLSM type equations describe the evolution of the electromagnetic field in the quadratically polarized media. Both scalar and vector NLSM systems, in three space + one time dimension, were obtained.

• Numerical calculations of NLSM type equations in case of nonlinear optics were carried out by Crasovan, Torres et al. (2003) Indications of wave collapse were found in certain parameter regime.


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The electric polarization field of intense laser beams propagating in optical mediacan be expanded in powes of the electric field as

Derivation of NLSM in optics

(1) (2) (3) ...P E E E E E Eχ χ χ= × + × × + × × × +

1 2 3( , , )E E E E= :Electric field vector

( )jχ : Susceptibility tensor coefficients of the medium

Quasi monochromatic expansion of the component of the electromagnetic Field with the fundamental harmonic, second harmonic and a mean term is

x

1( ) 2 2 ( )

2~ [ . .] [ . . ] ...xi ikx t kx tE Ae c c A e c cω ωε ε φ− −+ + + + +

Using a polarization field of the form (*) in Maxwell’s equations leads to NLSM Type equations for non zero (2)χ

(*)


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• Ablowitz, Biondini and Blair (1997)

For scalar system, if the time dependence in these equations is neglectedand problem is reconsidered for the materials belong to a special symmetry class then it can be seen that these equations are NLSM type equations.

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,1 ,1 ,0[2 (1 ) ] 0Z x XX YY TT x x xik kk M A M Aα φ′′∂ + − ∂ + ∂ − ∂ + + =2

,0 ,0 ,1 ,2[(1 ) ] [ ]( )x XX YY x TT x y XY y x TT x XXs N N Aα φ α φ− ∂ + ∂ + ∂ − ∂ = ∂ − ∂

In optics, U is the normalized amplitude of the envelope of the optical beam andV is the normalized static field, ρ is the coupling constant which comes from the combined optical rectification- electro optic effect and is the asymmetryparameter comes from the anisotropy of the material. This system is recentlyInvestigated by Crasovan et al.(2003)

ν

02

1 2 =−+∆+ UVUUUiU z ρ

xxyyxx UVV )(2=+ν


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IntegribilityIntegribility of NLSMof NLSM

1- When derivatives with respect to y can be neglected (e.g., in a narrow canal)the second equation can be integrated immediately, and one recovers the one-dimensional nonlinear Schrödinger equation which can be solved by the inversescattering transform (IST). M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (1981)

2. In deep water limit, the mean flow vanishes and NLSM equations reduce to(2+1)-dimensional NLS equation:

2iA A A A Aτ ξξ ηηλ µ χ∞ ∞ ∞+ + =

Contrary to the one-dimensional case, this equation is likely not solvable by IST. Also, for various choices of parameters the solutions can blow up in finite time.


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3- A different scenario arises in the opposite limit,that is shallow water.

In this case, after rescaling, the equations can be written as :

2

t xx yy xiA A A A A Aσ σ− + = + Φ22( )xx yy xAσ Φ + Φ = −

with 1σ = + 1σ = −or

• This system, usually called the Davey-Stewartson (DS I or DS II) equations, is of IST type, and thus completely integrable.

• For the Davey-Stewartson system, several exact solutions are available.In particular, stable localized pulses, often called dromions are knownto exist.

• Existence and well-posedness of solutions to NLSM type equations wasstudied by Ghidaglia and Saut (1990).

• Behavior of the blow up singularity was analyzed by Papanicolaou (1994).


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Global existence and collapse for NLSM

2

0( ) ( )N u u N u= =∫2 4 2 21 1

( , ) ( )2 2 2NLSM x yH u u u

ρφ φ νφ= ∇ − + +∫ ∫ ∫Hamiltonian

Virial Theorem holds

2 22 22 ( ) 4 ,NLSM

dx y u H

dz+ =∫

Papanicolaou et al. (1994)

Thus, in optics case, the coupling to the mean field corresponds to a self-defocusing mechanism, while in water waves case, it corresponds to a self-focusing mechanism => focusing in water waves case is easier to attain.

Power


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NLS Ground State

zieyxRu λ),(=

02

1 3 =+∆+− RRRλ

∫= 2RNc

NLS stationary solutions, which are obtained by substituting into the NLS equation, satisfy

The ground state of the NLS can be defined as a solution in H1 of this equation having the minimal power of all the nontrivial solutions. The existence and uniqueness of the ground state have been proven. Ground state is radially symmetric, positive and monotonically decaying.

cNN <Solution exists globally for where


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NLSM Ground State

),(,),( yxGeyxFu zi == φλ

02

1 3 =−+∆+− xFGFFF ρλ

∫= ),;,(),( 2 ρνρν yxFNc

NLSM stationary solutions, which are obtained by substituting into the NLSM equation, satisfy

The ground state of the NLSM can be defined as a nontrival solution (F, G) in H1 such that Fhas the minimal power of all the nontrivial solutions. The existence of the ground state hasbeen proven by Cipolatti (92). In the same spirit as for NLS, Papanicolaou et al. (94) extendedthe global existence theory to the NLSM and proved that

cNN <Solution exists globally for where

xyyxx FGG )( 2=+ν

∫ ∫∫ =∇+−∇= 0)(22

1)(

2

1),( 242 GFFGFH NLSM ν

ρ 222)( yx GGG νν +=∇where


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• Investigating the blow up structure of NLSM type equations for bothoptics and water waves problem, in the context of :

♦ Hamiltonian approach which was introduced by Ablowitz andSegur (79)

♦ Global existence theory

♦ Numerical methods

• Obtaining the ground state mode :

AIM OF THE STUDY

( , )exp( )u F x y i zλ=


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NumericalNumerical methodmethod & & InitialInitial ConditionsConditions forfor OpticsOptics andand

WaterWater WavesWaves CasesCases

• Ground state mode is obtained by using a fixed point numericalprocedure similar to what was used by Ablowitz and Musslimani(2003) in dispersion-managed soliton theory.

• For Hamiltonian approach and direct simulation, a symmetricGaussian type of inital condition is used

2 2( )0

2( , , 0)G x yN

u x y z eπ

− += = ( )N N G=where

Hamiltonian

2

0 0( , ) 121

G G NH u N

ρφπν

= − − +

is the input power


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Threshold power for which H=0 , given by2

( , )1 /(1 )

HcN

πν ρρ ν

≡− +

Such that when then and, therefore, the solution collapses at finite distance.

HCN N> 0H <

Alternatively,

2( , ) 1 (1 )H

c NN

πρ ν ν = − +

Such that when then and collapse is guaranteed by the Virial Theorem.

Hcρ ρ< 0H <


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Critical power for collapse as a function of forρ 0.5ν =

N<Nc

H<0


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The regions in the corresponding to collapse and glo bal-existence

(a) Nonlinear optics (b) Water waves

N<Nc

H<0H<0

N<Nc


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NLSMNLSM MODEMODE( , ) (0.2,0.2)ν ρ =


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The on-axis amplitudes of the ground state & Contou r plots

0.5ν =For

NLS TOWNES

OPTICS

opticsWater waves


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The astigmatism of the ground state F(x,y)

(a) ν = 0.5 with -1 ≤ ρ ≤ 1 (b) ρ = -0.2 (dashes) and ρ= 0.2(solid)with 0 ≤ ν ≤ 1

2

2

( )( )

( )

y

x

ue z

u= ∫∫


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Input Astigmatism~Astigmatic initial conditions

[ ]22)(0

2),( yExE e

ENyxu +−=

π axisyalongElongationE

axisxalongElongationE

SymmetryRadialE

1

10

1

><<

=

πνρφ

2/11

2

1),(

22

00

EN

EN

EuH EE

+−−+=

)/1/(1

)/1(

E

EEN H

c νρπ

+−+=

∞=→−= Hcc NE )1/(ρν

As input beam becomes narrower along the x-axis, the critical power for collapse increases, making the collapse more difficult to attain.

For optics case:


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(a) The focusing factor of the NLSM solutions (b) The corresponding astigmatism of the solution as a function of the

focusing factor(Input power is taken as N=1.2 Nc(ν = 0.5, ρ = -1)≈12.2)

OPTICSNLS TOWNES

WATER WAVES


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In order to study the self-similarity of the collapse process, the modulationfunction is recovered from the solution as

(0,0)( )

(0,0, )F

L zu z

=

The rescaled amplitude of the solution of the NLSM, i.e ( , , )L u Lx Ly z% %

is compared with ground state and ( , )F x y%% ( )( , ) / , /x y x L y L=%%

In order to show that the collapse process is quasi-self similar with the corresponding ground state, the rescaled amplitude is shown to converge

pointwise to as F cz Z→

Self-similarity of the collapse profile


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Convergence of the modulated collapse profile (dashes) to the NLSM ground state (solid)

Along x axis (top) and along y axis (bottom) with (ν, ρ) = (0.5,1)


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Along x axis (top) and along y axis (bottom) with (ν, ρ) = (0.5,-1)


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Along x axis (top) and along y axis (bottom) with (ν, ρ) = (4,- 4)


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Along x axis (top) and along y axis (bottom) with (ν, ρ) = (4,- 4)(semi-log plot)


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Collapse Arrest

012

12

2

=+

−+∆+

u

uuuuiu x

z εφρ

xyyxx u )(2=+νφφ


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Related NLSM Type System

02

1 =−∆+ xz uuiu φρ

xyyxx u )(2=+νφφ

∫ ∫ ++∇= )(22

1),( 222

yxuuH νφφρφ

Consider the NLSM system without the cubic term

Hamiltonian

Virial Theorem is not changed and collapse is possible for negative ρ

Substituting the initial conditions into the Hamiltonian, the threshold power for zero Hamiltonian

ρνπρν )1(2

),(+−=H

cN


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CONCLUSIONSCONCLUSIONS• Direct numerical simulation results are consistent with the Virial Theorem and Global

Existence Theory. This is in the same spirit as the results of classical NLS equation.

• In contrast to the NLS case, stationary solutions of NLSM are not radially symmetric but elliptic.

• Ground state profile is astigmatic and therefore, the collapse profile is astigmatic.

• The singularity occurs in water waves more quickly than in optics.

• As z approaches to zc (collapse distance) numerical simulations show that the nature of singularity for both optics and water waves, is described by a self-similar collapse profile given in terms of the ground state profile.

• From the experimental perspective, self similar collapse in quadratic-cubic media remains to be demonstrated in either free-surface waves and nonlinear optics.

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Wave Collapse in Nonlocal Nonlinear Schrödinger …Nonlinear Physics. Theory and Experiment IV 2006...

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