Existence and orbital stability of standing waves for the 1D Schr\"odinger-Kirchhoff equation

TL;DR

This paper proves the orbital stability of standing wave solutions for the 1D Schr"odinger-Kirchhoff equation, highlighting how the mixed term affects dispersion and stability compared to the nonlinear Schr"odinger equation.

Contribution

It establishes the orbital stability of standing waves for the 1D Schr"odinger-Kirchhoff equation and provides explicit solutions for periodic waves.

Findings

Orbital stability of standing waves is proven.

Explicit solutions for periodic waves are constructed.

The mixed term influences dispersion and stability behavior.

Abstract

In this paper we establish the orbital stability of standing wave solutions associated to the one-dimensional Schr\"odinger-Kirchhoff equation. The presence of a mixed term gives us more dispersion, and consequently, a different scenario for the stability of solitary waves in contrast with the corresponding nonlinear Schr\"odinger equation. For periodic waves, we exhibit two explicit solutions and prove the orbital stability in the energy space.