Variational methods for breather solutions of nonlinear wave equations

Rainer Mandel,Dominic Scheider

doi:10.1088/1361-6544/abed38

Abstract

We construct infinitely many real-valued, time-periodic breather solutions of the nonlinear wave equation with suitable N ⩾ 2, p > 2 and localized nonnegative Q. These solutions are obtained from critical points of a dual functional and they are weakly localized in space. Our abstract framework allows to find similar existence results for the nonlinear Klein–Gordon equation and biharmonic wave equations.

Highlights

Breathers are real-valued, time-periodic and spatially localized solutions of nonlinear equations describing the propagation of waves on RN ×R where N ∈ N
The existence of breather solutions appears to be rare phenomenon and up to now, most work in this area is related to the discussion of explicit examples such as the famous sine-Gordon breather for the (1 + 1)sine-Gordon equation [1]
In this paper we propose a variational approach for the construction of weakly localized breather solutions that does not rely on symmetry assumptions for the coefficients and generalizes to much more general classes of equations

Summary

Introduction

Breathers are real-valued, time-periodic and spatially localized solutions of nonlinear equations describing the propagation of waves on RN ×R where N ∈ N. The first deals with a semilinear curl-curl wave equation in R3 × R where −uxx is replaced by ∇ × ∇ × u in (1) and u is a three-dimensional vector field on R3 Using that this part in the equation vanishes for gradient fields, Plum and Reichel [28] succeed in proving the existence of exponentially localized breather solutions via ODE methods for suitable radially symmetric coefficient functions s, q and power-type nonlinearities f. In this paper we propose a variational approach for the construction of weakly localized breather solutions that does not rely on symmetry assumptions for the coefficients and generalizes to much more general classes of equations.

Applications and Examples

Proof of Theorem 1

Proofs of auxiliary results