# Journal of Nonlinear Mathematical Physics

Volume 22, Issue 3, June 2015, Pages 448 - 473

# Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

Authors
Spyridon Kamvissis
Department of Applied Mathematics, University of Crete Knossos-Heraklion, 71409, Crete, Greece.spyros@tem.uoc.gr
Dmitry Shepelsky
B. Verkin Institute for Low Temperature Physics and Engineering 47 Lenin Avenue, 61103 Kharkiv, Ukraine.shepelsky@yahoo.com
Lech Zielinski
Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral Côte d'Opale, 50 rue Ferdinand Buisson - B.P. 699 62228 Calais, France.Lech.Zielinski@lmpa.univ-littoral.fr
Received 13 February 2015, Accepted 19 June 2015, Available Online 6 January 2021.
DOI
10.1080/14029251.2015.1079428How to use a DOI?
Keywords
initial boundary value problems; oscillatory initial data; nonlinear Schrödinger equation
Abstract

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(−2iβx) (or asymptotically periodic, u(x, 0) =α exp(−2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: ux(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0.

Open Access

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
22 - 3
Pages
448 - 473
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2015.1079428How to use a DOI?
Open Access

TY  - JOUR
AU  - Spyridon Kamvissis
AU  - Dmitry Shepelsky
AU  - Lech Zielinski
PY  - 2021
DA  - 2021/01/06
TI  - Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 448
EP  - 473
VL  - 22
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2015.1079428
DO  - 10.1080/14029251.2015.1079428
ID  - Kamvissis2021
ER  -