Existence of ground states for a one-dimensional relativistic Schrödinger equation

Abstract

Relativistic Schrödinger equation with a nonlinear potential interaction describes the dynamics of a particle, with rest mass m, travelling to a significant fraction |v| < 1 of the light speed c = 1. At first, we deal with the local and global existence of solutions of the flux, and in the second term, and according to the relativistic nature of the problem, we look for boosted solitons as ψ(x, t) = eiμtφv(x − vt), where the profile \documentclass[12pt]{minimal}\begin{document}$\varphi _{v}\in H^{1/2}\left( \mathbb {R}\right)$\end{document}φv∈H1/2R is a minimizer of a suitable variational problem. Our proof uses a concentration-compactness-type argument. Stability results for the boosted solitons are established.