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Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearity

Xin Meng, Shuguan Ji

Abstract

This paper is concerned with breather solutions of a radially symmetric curl-curl wave equation with double power nonlinearity. By considering the solutions with a special form, we obtain a family of ordinary differential equations (ODEs) parameterized by the radial variable. Then we characterize periodic behaviors and analyze the joint effects of the double power nonlinear terms on the minimal period and the maximal amplitude. Under certain conditions, we construct a breather solution for the original curl-curl wave equation and find such a solution which can generate a continuum of phase-shifted breathers.

Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearity

Abstract

This paper is concerned with breather solutions of a radially symmetric curl-curl wave equation with double power nonlinearity. By considering the solutions with a special form, we obtain a family of ordinary differential equations (ODEs) parameterized by the radial variable. Then we characterize periodic behaviors and analyze the joint effects of the double power nonlinear terms on the minimal period and the maximal amplitude. Under certain conditions, we construct a breather solution for the original curl-curl wave equation and find such a solution which can generate a continuum of phase-shifted breathers.
Paper Structure (5 sections, 6 theorems, 51 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Qualitative analysis
  4. Main results
  5. Conclusion

Key Result

Lemma 2.1

Let $\psi:[0,\infty)\rightarrow\mathbb{R}$ be a $C^2$-function and let $F:\mathbb{R}^3\setminus\left\{0\right\}\rightarrow \mathbb{R}^3$ be given by $F(x):=\psi(|x|)\frac{x}{|x|}$. Then $F$ can be extended to a function (i) $F\in C^1(\mathbb{R}^3)$ if and only if $\psi(0)=0$; (ii) $F\in C^2(\mathbb{

Figures (2)

  • Figure 1: Part of the phase plane of \ref{['eq1.4']} for p=3 and q=4.
  • Figure 2: (a) and (b) show part of the breather solution for p=3 and q=4.

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • ...and 4 more