On decay properties of solutions of the $k$-generalized KdV equations

TL;DR

This paper investigates decay properties of solutions to the $k$-generalized KdV equations, focusing on persistence in weighted Sobolev spaces, unique continuation, and decay of perturbations of traveling waves.

Contribution

It establishes new decay and unique continuation properties for solutions, extending understanding of solution behavior in weighted Sobolev spaces.

Findings

Solutions exhibit specific decay properties in weighted Sobolev spaces.

Unique continuation properties are characterized sharply.

Decay behavior of perturbations of traveling waves is analyzed.

Abstract

We prove special decay properties of solutions to the initial value problem associated to the $k$-generalized Korteweg-de Vries equation. These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.