Spatial decay of multi-solitons of the generalized Korteweg-de Vries and nonlinear Schr{\"o}dinger equations

TL;DR

This paper investigates the spatial decay properties of multi-soliton solutions in generalized Korteweg-de Vries and nonlinear Schr{"o}dinger equations, demonstrating exponential decay in specific regions and rapid decay elsewhere.

Contribution

It provides new rigorous results on the uniform spatial decay rates of multi-solitons for these equations, including derivatives, in both KdV and NLS contexts.

Findings

Exponential decay of solutions and derivatives on the left of solitons

Rapid decay of solutions outside the solitons region

Uniform decay estimates in space for multi-solitons

Abstract

We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schr{\"o}dinger equations, that is, exponential decay in the solitons region and rapid decay outside.