Solutions of the Nonlinear Schrodinger Equation with Prescribed Asymptotics at Infinity

Details Solutions of the Nonlinear Schrodinger Equation with Prescribed Asymptotics at Infinity Solutions of the Nonlinear Schrodinger Equation with Prescribed Asymptotics at Infinity

2009-09-17

arxiv.org/abs/0909.3141v3

Mathematics

Analysis of PDEs

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

Scientific paper

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.

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