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On the steadiness of symmetric solutions to higher order perturbations of KdV

Long Pei, Fengyang Xiao, Pan Zhang

TL;DR

The paper investigates symmetry constraints on traveling waves in two higher-order perturbations of the KdV equation relevant to shallow-water waves: the generalized Rosenau-Kawahara-RLW equation and a perturbed R-KdV-RLW model. Using both classical and weak formulations, it shows that any spatially symmetric solution to the generalized equation is a traveling wave with speed tied to the symmetry axis, and it derives explicit conditions under which traveling solutions exist. For the more complex perturbed model, it proves there is no nontrivial symmetric solitary traveling solution when dissipation or shoaling terms are active, while also providing a constructive, albeit case-specific, example of symmetric traveling solutions under particular coefficients and initial data. These results offer a rigorous criterion to assess the suitability of higher-order KdV-type models for water waves and demonstrate the sharpness of symmetry-based arguments in dispersive PDEs.

Abstract

We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitableness of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden \& Raynaud] for solitary waves.

On the steadiness of symmetric solutions to higher order perturbations of KdV

TL;DR

The paper investigates symmetry constraints on traveling waves in two higher-order perturbations of the KdV equation relevant to shallow-water waves: the generalized Rosenau-Kawahara-RLW equation and a perturbed R-KdV-RLW model. Using both classical and weak formulations, it shows that any spatially symmetric solution to the generalized equation is a traveling wave with speed tied to the symmetry axis, and it derives explicit conditions under which traveling solutions exist. For the more complex perturbed model, it proves there is no nontrivial symmetric solitary traveling solution when dissipation or shoaling terms are active, while also providing a constructive, albeit case-specific, example of symmetric traveling solutions under particular coefficients and initial data. These results offer a rigorous criterion to assess the suitability of higher-order KdV-type models for water waves and demonstrate the sharpness of symmetry-based arguments in dispersive PDEs.

Abstract

We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitableness of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden \& Raynaud] for solitary waves.

Paper Structure

This paper contains 9 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Classical symmetric solutions of Rosenau type equations
  4. The generalized Rosenau-Kawahara-RLW equation
  5. Specific models in the Rosenau-Kawahara-RLW family
  6. Symmetric solutions in weak formulation for the generalized Rosenau-Kawahara-RLW
  7. Classification of symmetric steady solutions of the R-KDV-RLW equation
  8. Non-existence of symmetric steady solutions of solitary type
  9. A admissible set of coefficients and initial data to guarantee nontrivial symmetric traveling solutions

Key Result

Lemma 3.1

Assume $s\geq0$. Let $u_{0}\in H^{s}(\mathbb{R})$. Then, there exists $T>0$ and a unique solution $v \in X^{s}:=C(I,H^{s}(\mathbb{R}))$ for initial value $v_{0}\in H^{s}(\mathbb{R})$, where the norm on $X^{s}$ is defined as

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1: Well-posedness for Rosenau-Kawahara-RLW
  • Theorem 3.2
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 4 more