Numerical study of the semiclassical limit of the Davey–Stewartson II equations

McLaughlin

et al. 2019

Comm Pure Appl Math

The defocusing Davey‐Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one‐dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space‐time that become well‐defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey‐Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB‐type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k‐values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one‐dimensional theory. © 2019 Wiley Periodicals, Inc.

Section: Introductionmentioning

confidence: 91%

A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

Assainova

McLaughlin

et al. 2019

Comm Pure Appl Math

“…If one is interested in the solution to the DS II equation for localized initial data varying on length scales of order

1 / ε

, and this for times of order

1 / ε

with

ε ≪ 1

, one way to treat this is a change of coordinates

x \mapsto ε x

y \mapsto ε y

, and

t \mapsto ε t

. For , this leads to the equation (in an abuse of notation we use the same symbols as before)

\begin{matrix} i ε \partial_{t} ψ + ε^{2} \partial_{x x} ψ - ε^{2} \partial_{y y} ψ - 2 (() Φ + {| ψ |}^{2}) ψ & = & 0, \\ \partial_{x x} normalΦ + \partial_{y y} normalΦ + 2 \partial_{x x} {false| ψ false|}^{2} & = & 0 . \end{matrix}

Thus, it is possible to consider a family of equations (depending on the parameter ε) for ε‐dependent initial data, which allows us to study the semiclassical limit of DS II (see for instance ).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In , this approach was discussed with considerably higher resolution in detail and used to quantitatively identify the time where the distance δ of the singularity from the real axis becomes smaller than the minimal resolved distance via Fourier methods, i.e.,

m : = 2 π D / N

with

N \in N

being the number of Fourier modes and

2 π D

the length of the computational domain in physical space. A value of

δ < m

cannot be distinguished numerically from 0.…”

Section: Methodsmentioning

confidence: 99%

Numerical Study of Blow‐Up Mechanisms for Davey–Stewartson II Systems

Stoilov

2018

Stud Appl Math

We present a detailed numerical study of various blow‐up issues in the context of the focusing Davey–Stewartson II equation. To this end, we study Gaussian initial data and perturbations of the lump and the explicit blow‐up solution due to Ozawa. Based on the numerical results it is conjectured that the blow‐up in all cases is self‐similar, and that the time‐dependent scaling behaves as in the Ozawa solution and not as in the stable blow‐up of standard L2 critical nonlinear Schrödinger equation. The blow‐up profile is given by a dynamically rescaled lump.

“…To study the solution for the potential (23) for = 1/128 and k = 1, we use N x = 2 9 and N y = 2 10 Fourier modes and L x = 2, L y = 1. The solution can be seen in Fig.…”

Section: Solution Via Gmresmentioning

confidence: 99%

Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation

Physica D: Nonlinear Phenomena

McLaughlin²,

Stoilov

2019

The inverse scattering approach for the defocusing Davey-Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.