Wave Collapse in Nonlocal Nonlinear Schrdinger Equations

1
Wave Collapse in Nonlocal Nonlinear Schrödinger
Equations

• I. BAKIRTAS
• ITÜ 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
2
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
  (21)D focusing cubic nonlinear Schrödinger
  equation NLS

3
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)

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 .
4

• Subsequently many researchers have studied the
  NLS in detail
• Weinstein (1983) showed that when the power is
  sufficiently small, i.e.,

The solution exists globally.
Therefore, the sufficient condition for collapse
is
While the necessary condition for collapse is
Weinstein also found that the ground state of the
NLS also plays an important role in the collapse
theory. The ground state is a stationary
solution of the form
5

• 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.

6

• There are considerably fewer studies of the
  wave collapse that arise in nonlinear media whose
  governing equations have quadratic
  nonlinearities,
• 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.

7
For the physical models considered in this study,
the general nonlinear Schrödinger-mean (NLSM)
system can be written in the following form

• 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

Which corresponds to a strongly-nonlocal function
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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)

9
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
direction of propagation
transverse direction
time
measure of the weak nonlinearity
coefficients of the zeroth, first, second
harmonics
Substituting the wave expansion into Eulers
equations with a free surface and assuming slow
modulations of the field in and
directions results a nonlinearly coupled system
for and .
10

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

where
Dimensionless coord.,
are the wave numbers in the
directions,
group velocity
are suitable functions of
wave number, dispersion coefficients
and surface tension
where
normalized water depth
11
By rescaling the variables, previous system can
be transformed to
(Elliptic-elliptic case), this system admits
Collapse, requires large surface tension
For
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Hamiltonian Virial Theorem

• AblowitzSegur (1979) defined the Hamiltonian

Each bracket, , in H is positive definite, and
the second bracket vanishes
in the linear limit of Benney Roskes equations.
Clearly Hlt0 is possible.
Furthermore, they showed that the Virial Theorem
holds
As can be seen if H lt0, 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.
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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
  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
  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.

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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
()
Electric field vector
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
Using a polarization field of the form () in
Maxwells equations leads to NLSM Type equations
for non zero
15

• 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.

In optics, U is the normalized amplitude of the
envelope of the optical beam and V is the
normalized static field, ? 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)
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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 one- dimensional 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 (21)-dimensional NLS equation
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
with
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 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).

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Global existence and collapse for NLSM
Papanicolaou et al. (1994)
Power
Hamiltonian
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 gt
focusing in water waves case is easier to attain.
Virial Theorem holds
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NLS Ground State
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.
Solution exists globally for
where
20
NLSM Ground State
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 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
where
where
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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

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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

is the input power
where
Hamiltonian
23
Threshold power for which H0 , given by
Such that when then
and, therefore, the solution collapses at
finite distance.
Alternatively,
Such that when then
and collapse is guaranteed by the Virial
Theorem.
24
Critical power for collapse as a function of
for
Hlt0
NltNc
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The regions in the corresponding to collapse and
global-existence
NltNc
NltNc
Hlt0
Hlt0
(a) Nonlinear optics
(b) Water waves
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NLSM MODE
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The on-axis amplitudes of the ground state
Contour plots
OPTICS
NLS TOWNES
optics
Water waves
For
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The astigmatism of the ground state F(x,y)

• (a) ? 0.5 with -1 ? 1
  (b) ? -0.2 (dashes) and ? 0.2(solid)
• 
  with 0 ? 1

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Input AstigmatismAstigmatic initial conditions
For optics case
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
OPTICS
NLS TOWNES

• The focusing factor of the NLSM solutions
• The corresponding astigmatism of the solution as
  a function of the focusing factor
• (Input power is taken as N1.2 Nc(?
  0.5, ? -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
The rescaled amplitude of the solution of the
NLSM, i.e
is compared with ground state and

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
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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 (?, ?) (0.5,1)

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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
  (?, ?) (0.5,-1)

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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
  (?, ?) (4,- 4)

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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
  (?, ?) (4,- 4)
• (semi-log plot)

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Collapse Arrest
37
Related NLSM Type System
Consider the NLSM system without the cubic term
Hamiltonian
Virial Theorem is not changed and collapse is
possible for negative
Substituting the initial conditions into the
Hamiltonian, the threshold power for zero
Hamiltonian
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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.