The modified Korteweg-de Vries equation on the quarter plane with t-periodic data
- 10.1080/14029251.2017.1375695How to use a DOI?
- Initial-boundary value problem; Integrable systems; Modified Korteweg-de Vries equation
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 ux(0,t) and uxx(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.
- © 2017 The Authors. Published by Atlantis Press and Taylor & Francis
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Cite this article
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 -