Periodic soliton solutions to nonlinear Klein-Gordon equations on \({\mathbb{R}}^ 2\)
The paper deals with nonlinear one-dimensional Klein-Gordon equations of the form \[ (1)\quad u_{tt}(x,t)=u_{xx}(x,t)-g(u(x,t)) \] where \((x,t)\in {\mathbb{R}}^ 2\), g: \({\mathbb{R}}\to {\mathbb{R}}\), \(g(u_ 0)=0\) and \(g'(u_ 0)>0\) for some \(u_ 0\in {\mathbb{R}}\). The author proves the existence and the regularity of time-periodic spatially-decaying towards \(u_ 0\) (as \(x\to \pm \infty)\) small-amplitude solutions to equations of the form (1) restricted to the half-spaces \({\mathbb{R}}_ 0^{\pm}\times {\mathbb{R}}\). The author also presents nonexistence results for small- amplitude homoclinic orbits of a certain dynamical system associated with equation (1) on \({\mathbb{R}}^ 2\), which demonstrate the generic impossibility to continue the above-mentioned solutions defined on \({\mathbb{R}}_ 0^{\pm}\times {\mathbb{R}}\) to the whole of \({\mathbb{R}}^ 2\). The obtained existence and nonexistence results are applied to several semilinear wave equations of current interest in theoretical physics.
scientific article; zbMATH DE number 4208670