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

DOI

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 g̃₀ (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 u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g̃₁ (t) and g̃₂ (t) as t → ∞, respectively, then the periodic functions g̃₁ (t) and g̃₂ (t) can be uniquely determined in terms of the function g̃₀ (t). Furthermore, we characterize the Fourier coefficients of g̃₁ (t) and g̃₂ (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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TY  - JOUR
AU  - Guenbo Hwang
PY  - 2021
DA  - 2021/01/06
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  - 10.1080/14029251.2017.1375695
ID  - Hwang2021
ER  -