Breathers for a relativistic nonlinear wave equation
The authors of this interesting paper study real-valued solutions to a relativistic nonlinear wave equation of the type \[ \partial^2 u/\partial t^2(x,t)=\partial^2 u/\partial x^2(x,t)+F(u(x,t)), \] \(x\in \mathbb R\). It is assumed that \(u\in C(\mathbb R ^2)\) and the nonlinear term \(F(u)\) is a distribution of the type \[ F(u)= \sum\limits_{k\in\mathbb Z}F_k\delta (u-z_k), \] where \(u\in \mathbb R\), \(\mathbb Z\) is a discrete closed subset of \(\mathbb R\), \(\delta (u-z_k) \) is \(\delta \)-function of Dirac. Then there exists a piecewise constant potential \(V(u)\) such that \(F(u)=-V^{\prime }(u)\). The sense of the equation is defined according to the least-action principle. The energy and momentum conservation is established. The main results are the existence of time-periodic finite-energy solutions, the existence of global solutions and soliton-type asymptotics for a class of finite-energy initial data.
scientific article; zbMATH DE number 1922364