Journal of Nonlinear Mathematical Physics

Volume 20, Issue 1, April 2013, Pages 135 - 157

The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum

Authors
Guenbo Hwang
Department of Mathematics, Daegu University, Gyeonsan Gyeongbuk 712-714, Korea,ghwang@daegu.ac.kr
A. S. Fokas
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK,t.fokas@damtp.cam.ac.uk
Received 14 January 2013, Accepted 15 February 2013, Available Online 6 January 2021.
DOI
10.1080/14029251.2013.792492How to use a DOI?
Keywords
Initial-boundary value problem; Generalized Dirichlet to Neumann map; modified Korteweg-de Vries equation
Abstract

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem uniquely determined in terms of the initial datum q(x,0), as well as of the boundary values {q(0, t),qx(0, t),qxx(0, t)}. For the Dirichlet problem, it is necessary to characterize the unknown boundary values qx(0, t) and qxx(0, t) in terms of the given data q(x, 0) and q(0, t). It is shown here that in the particular case of a vanishing initial datum and of a sine wave as Dirichlet datum, qx(0, t) and qxx(0, t) can be computed explicitly at least up to third order in a perturbative expansion and that at least up to this order, these functions are asymptotically periodic for large t.

Copyright
© 2013 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
20 - 1
Pages
135 - 157
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2013.792492How to use a DOI?
Copyright
© 2013 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Guenbo Hwang
AU  - A. S. Fokas
PY  - 2021
DA  - 2021/01/06
TI  - The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum
JO  - Journal of Nonlinear Mathematical Physics
SP  - 135
EP  - 157
VL  - 20
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2013.792492
DO  - 10.1080/14029251.2013.792492
ID  - Hwang2021
ER  -