Uniform Adiabatic Limit of Benney type Systems

TL;DR

This paper proves that solutions of the cubic nonlinear Schrödinger equation can be obtained as asymptotic limits of solutions to Benney and related systems in one dimension, improving convergence conditions.

Contribution

It establishes uniform adiabatic limits for Benney type systems, including Zakharov and Zakharov-Rubenchik systems, with improved convergence conditions in one dimension.

Findings

Solutions of cubic nonlinear Schrödinger are limits of Benney systems.

Convergence in $L^2$ and energy space is achieved.

Improves previous results for Zakharov system without certain initial conditions.

Abstract

In this paper we show that solutions of the cubic nonlinear Schr\"odinger equation are asymptotic limit of solutions to the Benney system. Due to the special characteristic of the one-dimensional transport equation same result is obtained for solutions of the one-dimensional Zakharov and 1d-Zakharov-Rubenchik systems. Convergence is reached in the topology $L^2(\mathbb{R})\times L^2(\mathbb{R})$ and with an approximation in the energy space $H^1(\mathbb{R})\times L^2(\mathbb{R})$. In the case of the Zakharov system this is achieved without the condition $\partial_t n(x,0) \in \dot H^{-1}(\mathbb{R})$ for the wave component, improving previous results.