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On the steadiness of symmetric solutions to two dimensional dispersive models

Long Pei, Fengyang Xiao, Pan Zhang

Abstract

In this paper, we consider the steadiness of symmetric solutions to two dispersive models in shallow water and hyperelastic mechanics, respectively. These models are derived previously in the two-dimensional setting and can be viewed as the generalization of the Camassa-Holm and Kadomtsev-Petviashvili equations. For these two models, we prove that symmetry of classical solutions implies steadiness in the horizontal direction. We also confirm the such connection between symmetry and steadiness in weak formulation, which includes in particular the peaked solutions.

On the steadiness of symmetric solutions to two dimensional dispersive models

Abstract

In this paper, we consider the steadiness of symmetric solutions to two dispersive models in shallow water and hyperelastic mechanics, respectively. These models are derived previously in the two-dimensional setting and can be viewed as the generalization of the Camassa-Holm and Kadomtsev-Petviashvili equations. For these two models, we prove that symmetry of classical solutions implies steadiness in the horizontal direction. We also confirm the such connection between symmetry and steadiness in weak formulation, which includes in particular the peaked solutions.
Paper Structure (5 sections, 6 theorems, 41 equations)

This paper contains 5 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Steadiness of $x$-symmetric solutions to Camassa-Holm-Kadomtsev-Petviashvili equation
  4. Steadiness of $x$-symmetric solutions to the hyperelastic compressible plate model
  5. Steadiness of $x$-symmetric solutions in weak formulation

Key Result

Theorem 1

Assume that the Camassa-Holm-Kadomtsev-Petviashvili equation eq:chkp reformulation has a unique classical solution $u(t,x,y)$, $(t,x,y)\in I\times\mathbb{R}^{2}$, for given initial data $u_{0}(x,y)=u(0,x,y)$. If $u(t,x,y)$ is $x$-symmetric, then it is a steady solution in the $x$-direction with spee

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Definition 3
  • Lemma 1
  • ...and 7 more