Solutions of the Nonlinear Schrodinger Equation with Prescribed Asymptotics at Infinity

TL;DR

This paper establishes local existence and uniqueness of solutions to the one-dimensional nonlinear Schrödinger equation with prescribed asymptotic behavior at infinity, using discretization methods and extending techniques from KdV equations.

Contribution

It introduces a novel approach to solving NLS with specified asymptotics at infinity, including the difference between solutions and asymptotic expansions.

Findings

Proved local existence and uniqueness of solutions.

Demonstrated solutions differ from asymptotic expansions by Schwartz class functions.

Applied discretization methods to solve a generalized NLS equation.

Abstract

We prove local existence and uniqueness of solutions for the one-dimensional nonlinear Schr\"odinger (NLS) equations $iu_t + u_{xx} \pm |u|^2 u = 0$ in classes of smooth functions that admit an asymptotic expansion at infinity in decreasing powers of $x$. We show that an asymptotic solution differs from a genuine solution by a Schwartz class function which solves a generalized version of the NLS equation. The latter equation is solved by discretization methods. The proofs closely follow previous work done by the author and others on the Korteweg-De Vries (KdV) equation and the modified KdV equations.