Semiclassical soliton ensembles for the three-wave resonant interaction equations

TL;DR

This paper constructs semiclassical soliton ensembles for the three-wave resonant interaction equations using inverse-scattering and WKB analysis, revealing complex wave behaviors akin to dispersive shock waves despite the non-dispersive nature.

Contribution

It introduces a novel semiclassical soliton ensemble approach for the three-wave equations, enabling explicit asymptotic and numerical analysis of wave interactions.

Findings

Space-time partitioned into quiescent, slowly varying, and oscillatory regions.

Ensemble construction provides accurate initial data approximation in the semiclassical limit.

Wave behavior resembles dispersive shock phenomena despite non-dispersive system.

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.