Semiclassical soliton ensembles for the three-wave resonant interaction equations

Abstract

The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space-time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg-de Vries and nonlinear Schrodinger equations, although the physical mechanism must be different in the absence of dispersion.

82 pages, 14 figures

References in corpus (5)

• The Semiclassical Modified Nonlinear Schroedinger Equation I: Modulation Theory and Spectral Analysis
• On the Generation of Dispersive Shock Waves
• Semiclassical soliton ensembles for the three-wave resonant interaction equations
• The Gaussian semiclassical soliton ensemble and numerical methods for the focusing nonlinear Schrödinger equation
• A second look at the Gaussian semiclassical soliton ensemble for the focusing nonlinear Schrödinger equation

Cited by in corpus (4)

• Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation
• On the Generation of Dispersive Shock Waves
• Semiclassical soliton ensembles for the three-wave resonant interaction equations
• Universality near the gradient catastrophe point in the semiclassical sine-Gordon equation