Solutions of the Nonlinear Schrodinger Equation with Prescribed Asymptotics at Infinity

Abstract

We prove local existence and uniqueness of solutions for the one-dimensional nonlinear Schrödinger (NLS) equations $i{u}_t+u_{xx}\pm |u{|}^{2}u=0$ 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.

arXiv comment41 pages double spaced, accepted, added a few references, made some typographical changes, rewrote lemma A.1 part 6.

References in corpus (1)

• Travelling waves for the Gross-Pitaevskii equation II