Monotonicity of limit wave speed of periodic traveling wave solutions via Abelian integral

TL;DR

This paper proves the monotonicity of the limit wave speed for periodic traveling waves in a perturbed KdV equation using Abelian integrals, resolving open problems and providing a simpler proof method with supporting numerical simulations.

Contribution

It introduces a new method based on Abelian integrals to analyze wave speed monotonicity, solving open problems and simplifying previous proofs.

Findings

Confirmed the monotonicity of wave speed limit

Established at most one isolated periodic wave

Numerical results match theoretical predictions

Abstract

In this article, we investigate monotonicity of limit wave speed of periodic traveling wave solutions for a perturbed generalized KdV equation via Abelian integral. We have answered an open problem outlined by Yan et al. (2014) and the conjecture proposed by Ouyang et al. (2022). Geometric singular perturbation theory allows for the reduction of a three-dimensional dynamical system to a near-Hamiltonian planar system. Furthermore, utilizing the monotonic behavior of the ratio of Abelian integrals, we develop a method to show the existence of at most one isolated periodic traveling wave which is much simpler proof than that in Yan et al.(2014). Finally, we present numerical simulations that perfectly match the theoretical outcomes.