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Monotonicity of the periodic waves for the perturbed generalized defocusing mKdV equation

Lin Lu, Aiyong Chen, Xiaokai He

TL;DR

This work investigates the monotonicity of the limit wave speed $c_0(h)$ for periodic traveling waves in the perturbed generalized defocusing mKdV equation. It blends geometric singular perturbation theory with Abelian-integral analysis and an involution to establish the persistence and speed–energy relation of traveling waves under small perturbations. For every positive integer $n$, the limit speed $c_0(h)$ is shown to be increasing in energy $h$ (with $c_0(0)=1$), and the speed is expressible as $c_0(h)=1/(1- ilde B_n(h))$ with $ ilde B_n'(h)>0$, alongside explicit bounds in low-degree cases supported by numerics. Overall, the paper extends previous results for the defocusing mKdV to the generalized case and connects perturbation theory with Abelian-integral methods to characterize periodic wave speeds.

Abstract

In this paper, we study the existence of periodic waves for the perturbed generalized defocusing mKdV equation using the theory of geometric singular perturbation. By Abelian integral and involution operation, we prove that the limit wave speed c_0(h) is monotonic with respect to energy h,and the lower bound of the limit wave speed is found. These works extend the main result of Chen et al. (2018) to the generalized case. Some numerical simulations are conducted to verify the correctness of the theoretical analysis.

Monotonicity of the periodic waves for the perturbed generalized defocusing mKdV equation

TL;DR

This work investigates the monotonicity of the limit wave speed for periodic traveling waves in the perturbed generalized defocusing mKdV equation. It blends geometric singular perturbation theory with Abelian-integral analysis and an involution to establish the persistence and speed–energy relation of traveling waves under small perturbations. For every positive integer , the limit speed is shown to be increasing in energy (with ), and the speed is expressible as with , alongside explicit bounds in low-degree cases supported by numerics. Overall, the paper extends previous results for the defocusing mKdV to the generalized case and connects perturbation theory with Abelian-integral methods to characterize periodic wave speeds.

Abstract

In this paper, we study the existence of periodic waves for the perturbed generalized defocusing mKdV equation using the theory of geometric singular perturbation. By Abelian integral and involution operation, we prove that the limit wave speed c_0(h) is monotonic with respect to energy h,and the lower bound of the limit wave speed is found. These works extend the main result of Chen et al. (2018) to the generalized case. Some numerical simulations are conducted to verify the correctness of the theoretical analysis.
Paper Structure (4 sections, 7 theorems, 70 equations, 3 figures)

This paper contains 4 sections, 7 theorems, 70 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Perturbation analysis
  4. Analysis by Abelian integral and involution

Key Result

Theorem 2.1

For any positive integer $n$, let Then there exists $\epsilon_n^*>0$ such that for each $\epsilon \in (0,\epsilon_n^*)$ and $h\in(0,d_n),$ equation (1) has a traveling wave solution where $c=c(\epsilon,h)$, and $u(\epsilon,h,c,\tau)$ is a solution of equation (6). Furthermore, let $c_0(h)=\lim\limits_{\epsilon\rightarrow 0}c(\epsilon,h),$ then $c_0(h)$ satisfies $c_0^\prime(h)>0$ for $h \in (0,d

Figures (3)

  • Figure 1: The phase portraits of system (\ref{['8']}) .
  • Figure 2: The graph of image of $W$ and the involution $\eta$.
  • Figure 3: The graph of the limit wave speed $c_0(h)$ for $n=4.$

Theorems & Definitions (10)

  • Theorem 2.1
  • Lemma 3.1: YanW2014
  • Lemma 4.1: Patra2024
  • Lemma 4.2: Patra2024WeiM2025
  • Lemma 4.3: Christopher2007
  • Proposition 4.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof