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

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

2019 | O. ASSAINOVA, C. KLEIN, K. D. T.-R. MCLAUGHLIN, P. D. MILLER
This paper studies the defocusing Davey-Stewartson II (DS-II) equation in the semiclassical limit, focusing on the direct spectral transform. The DS-II equation, a two-dimensional nonlinear partial differential equation, exhibits behavior in the semiclassical limit that resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation. The paper introduces a WKB-type method for the direct spectral transform, proving its formal validity for large spectral parameters and providing numerical evidence of its accuracy. The method involves solving a singularly perturbed elliptic Dirac system and a nonlinear eikonal problem. The paper also analyzes the reflection coefficient for real radial potentials at k = 0, showing that it converges to a limiting function supported on the domain where the eikonal problem does not have a global solution. The study provides a rigorous semiclassical analysis of the solution, yielding an asymptotic formula for the reflection coefficient and suggesting an annular structure for the solution. Numerical examples are given for Gaussian and non-radial potentials, showing the behavior of the reflection coefficient as the semiclassical parameter ε approaches zero. The paper concludes with a conjecture about the accuracy of the WKB method and the behavior of the reflection coefficient in the semiclassical limit.This paper studies the defocusing Davey-Stewartson II (DS-II) equation in the semiclassical limit, focusing on the direct spectral transform. The DS-II equation, a two-dimensional nonlinear partial differential equation, exhibits behavior in the semiclassical limit that resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation. The paper introduces a WKB-type method for the direct spectral transform, proving its formal validity for large spectral parameters and providing numerical evidence of its accuracy. The method involves solving a singularly perturbed elliptic Dirac system and a nonlinear eikonal problem. The paper also analyzes the reflection coefficient for real radial potentials at k = 0, showing that it converges to a limiting function supported on the domain where the eikonal problem does not have a global solution. The study provides a rigorous semiclassical analysis of the solution, yielding an asymptotic formula for the reflection coefficient and suggesting an annular structure for the solution. Numerical examples are given for Gaussian and non-radial potentials, showing the behavior of the reflection coefficient as the semiclassical parameter ε approaches zero. The paper concludes with a conjecture about the accuracy of the WKB method and the behavior of the reflection coefficient in the semiclassical limit.
Reach us at info@study.space
[slides] Analysis%2C Second edition | StudySpace