The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg–de Vries equations

DOI

10.1080/14029251.2018.1494777How to use a DOI?

Keywords

Necessary and sufficient conditions; left- and right-reflection coefficients; time-evolution of scattering matrix; time-dependence of the reflection coefficients; soliton-solutions

Abstract

The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg–de Vries (KdV) equation with dominant surface tension are formulated. The necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients of the scattering problem are established. The time-dependence t, t > 0 of each element of the scattering matrix s(k, t) is found in respective sector of the k-spectral plane by expansion formulas which are constructed from the known initial and boundary conditions of the IBVP. Knowing the right-reflection coefficient calculated from the elements of s(k, t), we solve the Gelfand–Levitan–Marchenko (GLM) equation in the inverse problem. Then the solution of the IBVP is expressible through the solution of the GLM equation. The asymptotic behavior at infinity of time of the solution of the IBVP is shown

Copyright

© 2018 The Authors. Published by Atlantis 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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TY  - JOUR
AU  - Pham Loi Vu
PY  - 2021
DA  - 2021/01/06
TI  - The description of reflection coefficients of the scattering problems for finding solutions of the Korteweg–de Vries equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 399
EP  - 432
VL  - 25
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1494777
DO  - 10.1080/14029251.2018.1494777
ID  - Vu2021
ER  -