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