Perturbative and Exact Results on the Neumann Value for the Nonlinear Schr?dinger on the Half-line

Physics and Mathematics of Nonlinear Phenomena 2013

Fokas, A.S.^1,2 ; Lenells, J.^3

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom^1

Research Center of Mathematics, Academy of Athens, Athens 11527, Greece^2

Department of Mathematics and Center for Astrophysics, Space Physics and Engineering Research, Baylor University, One Bear Place #97328, Waco, TX 76798, United States^3

关键词: Asymptotic forms;    Dinger equation;    Initial and boundary conditions;    Neumann boundary;    Periodic boundary conditions;    Periodic function;    Perturbation expansions;    Unknown boundary;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/482/1/012015/pdf
DOI  :  10.1088/1742-6596/482/1/012015

来源: IOP

【 摘 要 】

The most challenging problem in the implementation of the so-called unified transform to the analysis of the nonlinear Schrodinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called linearizable boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large t, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large t behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given τ-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular τ-periodic functions, which includes {a exp(iωt)|a > 0, ω ≥ a2}, then the large t behavior of the Neumann value is given by a τ-periodic function which can be computed explicitly.