Journal of Nonlinear Mathematical Physics

Volume 24, Issue 4, September 2017, Pages 620 - 634

The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

Authors
Guenbo Hwang
Department of Mathematics and Institute of Basic Science, Daegu University, Gyeongsan Gyeongbuk 38453, Korea. ghwang@daegu.ac.kr
Received 20 June 2017, Accepted 17 July 2017, Available Online 6 January 2021.
DOI
https://doi.org/10.1080/14029251.2017.1375695How to use a DOI?
Keywords
Initial-boundary value problem, Integrable systems, Modified Korteweg-de Vries equation
Abstract

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function 0 (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data ux(0,t) and uxx(0,t) are asymptotically t-periodic with period τ which tend to the functions 1 (t) and 2 (t) as t → ∞, respectively, then the periodic functions 1 (t) and 2 (t) can be uniquely determined in terms of the function 0 (t). Furthermore, we characterize the Fourier coefficients of 1 (t) and 2 (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.

Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC license.

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
24 - 4
Pages
620 - 634
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1080/14029251.2017.1375695How to use a DOI?
Copyright
© 2017 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC license.

Cite this article

TY  - JOUR
AU  - Guenbo Hwang
PY  - 2021
DA  - 2021/01
TI  - The modified Korteweg-de Vries equation on the quarter plane with t-periodic data
JO  - Journal of Nonlinear Mathematical Physics
SP  - 620
EP  - 634
VL  - 24
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2017.1375695
DO  - https://doi.org/10.1080/14029251.2017.1375695
ID  - Hwang2021
ER  -