Solutions of the Nonlinear Schrodinger Equation with Prescribed\n Asymptotics at Infinity

2009/09/16 by John B. Gonzalez, Gonzalez, John B.
Mathematics · Physics and Astronomy · #35A01 #35A02 #35C20 #35Q55 #65M06 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0909.3141

openalex publication_date 2009/09/16 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We prove local existence and uniqueness of solutions for the one-dimensional\nnonlinear Schr "odinger (NLS) equations iu_t + u_xx \± |u|² u = 0 in\nclasses of smooth functions that admit an asymptotic expansion at infinity in\ndecreasing powers of x. We show that an asymptotic solution differs from a\ngenuine solution by a Schwartz class function which solves a generalized\nversion of the NLS equation. The latter equation is solved by discretization\nmethods. The proofs closely follow previous work done by the author and others\non the Korteweg-De Vries (KdV) equation and the modified KdV equations.\n

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