Gain of regularity for a coupled system of generalized nonlinear Schr\"odinger equations

TL;DR

This paper demonstrates that solutions to a coupled nonlinear Schrödinger system become smoother over time due to dispersive effects, given initial data with certain regularity and decay properties.

Contribution

It establishes a regularity gain result for solutions of a coupled nonlinear Schrödinger system, linking initial data properties to solution smoothness over time.

Findings

Solutions gain regularity over time due to dispersion.

Regularity gain depends on initial data regularity and decay.

Results applicable to physical models like polarized laser propagation.

Abstract

In this paper we study the smoothness properties of solutions to a one-dimensional coupled nonlinear Schr\"{o}dinger system equations that describes some physical phenomena such as propagation of polarized laser beams in birefringent Kerr medium in nonlinear optics. We show that the equations dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data $(u_{0},\,v_{0})$ possesses certain regularity and sufficient decay as $|x|\rightarrow\infty,$ then the solution $(u(t),\,v(t))$ will be smoother than $(u_{0},\,v_{0})$ for $0 < t \leq T$ where $T$ is the existence time of the solution.