A decay property of solutions to the k-generalized KdV equation

TL;DR

This paper establishes conditions under which solutions to the k-generalized KdV equation exhibit decay properties in weighted L^2 spaces, using fractional derivative inequalities.

Contribution

It introduces a novel application of Leibnitz rule type inequalities for fractional derivatives to analyze decay in solutions of the k-generalized KdV equation.

Findings

Solutions belong to weighted L^2 spaces under certain conditions.

Fractional derivative inequalities are effective in studying decay properties.

Provides a framework for analyzing decay in nonlinear dispersive equations.

Abstract

We use a Leibnitz rule type inequality for fractional derivatives to prove conditions under which a solution $u(x,t)$ of the k-generalized KdV equation is in the space $L^2(|x|^{2s}\,dx)$ for $s \in \mathbb R_{+}$.