Lower Bounds for Non-Trivial Traveling Wave Solutions of Equations of KdV Type

TL;DR

This paper establishes a lower bound condition for solutions of KdV-type equations, showing that solutions bounded above by rapidly decaying traveling waves must be trivial, regardless of wave speed or amplitude.

Contribution

It proves a new lower bound criterion for non-trivial solutions of KdV-type equations based on decay rates of traveling waves.

Findings

Solutions bounded by exponentially decaying traveling waves are zero.

No restrictions on the traveling wave's size or speed are needed.

The result applies broadly to KdV-type equations.

Abstract

We prove that if a solution of an equation of KdV type is bounded above by a traveling wave with an amplitude that decays faster than a given linear exponential then it must be zero. We assume no restrictions neither on the size nor in the direction of the speed of the traveling wave.