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Wave Collapse in Nonlocal
Nonlinear Schrödinger
Equations
İ. BAKIRTAŞ
İTÜ 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
Nonlinear Physics. Theory and
Experiment IV 2006
1
COLLAPSE
• 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 of
physics, 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
1
2
iuz ( x, y, z )  (u xx  u yy )  u u  0,
2
u ( x, y,0)  u0 ( x, y)
Nonlinear Physics. Theory and
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Nonlinear Schrödinger Equation &
Collapse
• 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
d2
2
2
( x  y ) u  4H ,
2 
dz
Hamiltonian:
H
1
2
4
(

u

u
0
0 )

2
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 H  0 .
Nonlinear Physics. Theory and
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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.,
N   u0  const  N c  1.8623
2
The solution exists globally.
Therefore, the sufficient condition for collapse is
While the necessary condition for collapse is
H 0
N  Nc
Weinstein also found that the ground state of the NLS also plays an
important role in the collapse theory.
iz
The ground state is a “stationary” solution of the form u  R(r )e
Nonlinear Physics. Theory and
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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.
Nonlinear Physics. Theory and
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There are considerably fewer studies of the wave collapse that arise
in nonlinear media whose governing equations have quadratic
nonlinearities,
(2)
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 nonlocalnonlinear coupling between a dynamic field that is associated with
the first harmonic and a static field associated with the mean term.
Nonlinear Physics. Theory and
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For the physical models considered in this study, the general nonlinear
Schrödinger-mean (NLSM) system can be written in the following form
1
2
iut   1uxx  u yy    2u u  ux  0
2
2
xx   yy  ( u ) x
These equations are also sometimes referred to as Benney-Roskes
or Davey-Stewartson type and are nonlocal because the second
equation can be solved for 


 G( x  x, y  y )


2
u ( x , y , z ) dxdy 
x
G( x, y)  (4 ) 1 log( x 2  y 2 / )
Which corresponds to a strongly-nonlocal function 
Nonlinear Physics. Theory and
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NLSM EQUATION FROM WATER WAVES
•
NLSM equations were originally obtained by Benney and Roskes (1969) in
their study of the instability of wave packets in multidimensional water wave
packets 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 and
Roskes to include the surface tension.
•
Ablowitz and Segur (1979) analyzed the Benney- Roskes equations and
showed 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 studied
by Ghidaglia and Saut (1990)
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Derivation of NLSM in water waves
Free-surface gravity-capillary water waves NLSM results from a weakly
nonlinear quasi-monchromatic expansion of velocity potential as
 ( x, y, t ) ~  [ Aei (kxt )  c.c.  ]   2[ A2e2i ( kxt )  c.c.]  ...
x
t
: direction of propagation
: time
, A, A2
y : transverse direction

: measure of the weak nonlinearity
: coefficients of the zeroth, first, second harmonics
Substituting the wave expansion into Euler’s equations with a free surface
and assuming slow modulations of the field in x and y directions results
a nonlinearly coupled system for A and  .
Nonlinear Physics. Theory and
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• In the context of water waves,Ablowitz and Segur (1979), studied the
NLSM (Benney-Roskes) Equations in the following form
iA   A   A   A A  1 A
2
       ( A )
2
where
   k ( x  cg t ),    ly,    2 ( gk )1/ 2 t
(k , l )
are the wave numbers in the
 ,  ,  , 1 ,  , 
( x, y )
Dimensionless coord.,
directions,
are suitable functions of : group velocity
cg
 2 / k 2 ,  2 / l 2
h and surface tension T , where  , 1  0
wave number, dispersion coefficients
normalized water depth
Nonlinear Physics. Theory and
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By rescaling the variables, previous system can be transformed to
1
2
iut   1uxx  u yy    2u u  ux  0
2
xx   yy  ( u ) x
2
For
1   2  1
 0
(Elliptic-elliptic case), this system admits
Collapse, requires large surface tension
Nonlinear Physics. Theory and
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Hamiltonian & Virial Theorem
• Ablowitz&Segur (1979) defined the Hamiltonian
2
2



1
1

A

A

 1
4
2
2
H    

( )  ( ) d d
  (   ) A 
 



 
 2
 
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.
Furthermore, they showed that the Virial Theorem holds
 2  2  2  2
   A d  d  8H
 2     
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
FOCUSING.
Nonlinear Physics. Theory and
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NLSM EQUATION FROM OPTICS
•
In isotropic (Kerr) media, where the nonlinear response of the material
depends cubically on the applied field, the dynamics of a quasimonochromatic 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 type
equations. 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.
Nonlinear Physics. Theory and
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Derivation of NLSM in optics
The electric polarization field of intense laser beams propagating in optical media
can be expanded in powes of the electric field as
P   (1)  E   (2)  E  E   (3)  E  E  E  ...
E  ( E1, E2 , E3 )
 ( j)
(*)
:Electric field vector
: Susceptibility tensor coefficients of the medium
Quasi monochromatic expansion of the x component of the electromagnetic
Field with the fundamental harmonic, second harmonic and a mean term is
E1 ~  [ Aei ( kxt )  c.c.]   2[ A2e2i ( kxt )  c.c.  x ]  ...
Using a polarization field of the form (*) in Maxwell’s equations leads to NLSM
Type equations for non zero  (2)
Nonlinear Physics. Theory and
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• Ablowitz, Biondini and Blair (1997)
For scalar system, if the time dependence in these equations is neglected
and problem is reconsidered for the materials belong to a special symmetry
class then it can be seen that these equations are NLSM type equations.
[2ik  Z  (1   x,1 ) XX  YY  kk TT  M x ,1 A  M x ,0x ] A  0
2
[(1   x ,0 ) XX  YY  sx TT ]x   y ,0 XY  y  [ N x ,1TT  N x ,2 XX ]( A )
2
1
2
iU z  U  U U  UV  0
2
2
Vxx V yy  ( U ) xx
In optics, U is the normalized amplitude of the envelope of the optical beam and
V is the normalized static field, ρ is the coupling constant which comes from
the combined optical rectification- electro optic effect and  is the asymmetry
parameter comes from the anisotropy of the material. This system is recently
Investigated by Crasovan et al.(2003)
Nonlinear Physics. Theory and
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Integribility of 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 onedimensional nonlinear Schrödinger equation which can be solved by the inverse
scattering 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:
iA   A   A    A A
2
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.
Nonlinear Physics. Theory and
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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 :
iAt   Axx  Ayy   A A  A x
2
  xx   yy  2( A ) x
2
with
  1
or
  1
• 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 known
to exist.
• Existence and well-posedness of solutions to NLSM type equations was
studied by Ghidaglia and Saut (1990).
• Behavior of the blow up singularity was analyzed by Papanicolaou (1994).
Nonlinear Physics. Theory and
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Global existence and collapse for NLSM
Papanicolaou et al. (1994)
Power
Hamiltonian
N (u )   u N (u0 )
2
1
1

2
4
H NLSM (u,  )   u   u   (x2   y2 )
2
2
2
Thus, in optics case, the coupling to the mean field corresponds to a selfdefocusing mechanism, while in water waves case, it corresponds to a selffocusing mechanism => focusing in water waves case is easier to attain.
Virial Theorem holds
2
d2
2
2
( x  y ) u  4 H NLSM ,
2 
dz
Nonlinear Physics. Theory and
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NLS Ground State
NLS stationary solutions, which are obtained by substituting u  R( x, y)eiz
into the NLS equation, satisfy
1
 R  R  R 3  0
2
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.
Solution exists globally for
N  Nc
where
Nc   R2
Nonlinear Physics. Theory and
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NLSM Ground State
NLSM stationary solutions, which are obtained by substituting
into the NLSM equation, satisfy
u  F ( x, y)eiz ,   G( x, y)
1
 F  F  F 3  FGx  0
2
Gxx Gyy  ( F 2 ) x
The ground state of the NLSM can be defined as a nontrival solution (F, G) in H1 such that F
has the minimal power of all the nontrivial solutions. The existence of the ground state has
been proven by Cipolatti (92). In the same spirit as for NLS, Papanicolaou et al. (94) extended
the global existence theory to the NLSM and proved that
Solution exists globally for
H NLSM ( F , G ) 
N  Nc
where
N c ( ,  )   F 2 ( x, y; ,  )
1
1

2
4
2
(

F
)

F

(

G
)
0

2
2
2
where
Nonlinear Physics. Theory and
Experiment IV 2006
( G) 2  Gx2 Gy2
20
AIM OF THE STUDY
• Investigating the blow up structure of NLSM type equations for both
optics and water waves problem, in the context of :
♦ Hamiltonian approach which was introduced by Ablowitz and
Segur (79)
♦ Global existence theory
♦ Numerical methods
• Obtaining the ground state mode :
u  F ( x, y ) exp(i z )
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Numerical method & Initial Conditions for Optics and
Water Waves Cases
• Ground state mode is obtained by using a fixed point numerical
procedure similar to what was used by Ablowitz and Musslimani
(2003) in dispersion-managed soliton theory.
• For Hamiltonian approach and direct simulation, a symmetric
Gaussian type of inital condition is used
u ( x, y, z  0) 
G
0
2N

Hamiltonian
e
 ( x2  y 2 )
where
N  N (G )
is the input power


G
G
H (u0 , 0 )  N  1 
 1 
Nonlinear Physics. Theory and
Experiment IV 2006
N

 2
2
22
2
Threshold power for which H=0 , given by N ( ,  ) 
1   /(1   )
H
c
N  N CH then H  0 and, therefore, the solution
Such that when
collapses at finite distance.
Alternatively,
2 

 ( N , )  1   (1   )
N 

H
c
  cH
Such that when
by the Virial Theorem.
then
H 0
Nonlinear Physics. Theory and
Experiment IV 2006
and collapse is guaranteed
23
Critical power for collapse as a function of  for   0.5
H<0
N<Nc
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The regions in the corresponding to collapse and global-existence
N<Nc
N<Nc
H<0
(a) Nonlinear optics
H<0
(b) Water waves
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NLSM MODE
( ,  )  (0.2,0.2)
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The on-axis amplitudes of the ground state & Contour plots
OPTICS
NLS TOWNES
Water waves
For
optics
  0.5
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 ( u )y
2
The astigmatism of the ground state F(x,y) e( z ) 
(a) ν = 0.5 with -1 ≤ ρ ≤ 1
 ( u )x
2
(b) ρ = -0.2 (dashes) and ρ= 0.2(solid)
with 0 ≤ ν ≤ 1
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Input Astigmatism~Astigmatic initial conditions
u0E ( x, y) 
2EN ( Ex)
e
2
y
2


E 1
0  E 1
Radial Symmetry
Elongation along x axis
E 1
Elongation along y axis
For optics case:
1 E2


 EN 2
H (u ,  ) 
N  1 

2
 1   / E  2
E
0
N cH 
E
0
( E  1 / E )
1   /(1   / E )
Ec   /(   1)  NcH  
As input beam becomes narrower along the x-axis, the critical power for collapse increases, making the
collapse more difficult to attain.
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WATER WAVES
NLS TOWNES
OPTICS
(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)
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Self-similarity of the collapse profile
In order to study the self-similarity of the collapse process, the modulation
function is recovered from the solution as
F (0,0)
L( z ) 
u (0,0, z )
The rescaled amplitude of the solution of the NLSM, i.e
is compared with
F ( x, y )
ground state and
L u ( Lx, Ly, z )
( 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
F
as
z  Zc
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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)
Nonlinear Physics. Theory and
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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)
Nonlinear Physics. Theory and
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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) = (4,- 4)
Nonlinear Physics. Theory and
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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) = (4,- 4)
(semi-log plot)
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Collapse Arrest
u u  u x
2
1
iu z  u 
2
2
1  u
0
 xx yy  ( u ) x
2
Nonlinear Physics. Theory and
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Related NLSM Type System
Consider the NLSM system without the cubic term
1
iu z  u  u x  0
2
 xx yy  ( u ) x
2
Hamiltonian
H (u,  ) 
1

2
2
2

u

(



)
x
y


2
2
Virial Theorem is not changed and collapse is possible for negative

Substituting the initial conditions into the Hamiltonian, the threshold power for zero Hamiltonian
N cH ( ,  )  
2 (1   )

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CONCLUSIONS
•
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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