Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T17:50:42.214Z Has data issue: false hasContentIssue false

THE mKdV EQUATION ON THE HALF-LINE

Published online by Cambridge University Press:  30 April 2004

A. Boutet de Monvel
Affiliation:
Institut de Mathématiques de Jussieu, Case 7012, Université de Paris 7, 2 Place Jussieu, 75251 Paris, France ([email protected])
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK ([email protected])
D. Shepelsky
Affiliation:
Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine ([email protected])

Abstract

An initial boundary-value problem for the modified Korteweg–de Vries equation on the half-line, $0<x<\infty$, $t>0$, is analysed by expressing the solution $q(x,t)$ in terms of the solution of a matrix Riemann–Hilbert (RH) problem in the complex $k$-plane. This RH problem has explicit $(x,t)$ dependence and it involves certain functions of $k$ referred to as the spectral functions. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of the boundary values $q(0,t)=g_0(t)$, $q_x(0,t)=g_1(t)$, and $q_{xx}(0,t)=g_2(t)$. The spectral functions satisfy an algebraic global relation which characterizes, say, $g_2(t)$ in terms of $\{q_0(x),g_0(t),g_1(t)\}$. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

AMS 2000 Mathematics subject classification: Primary 37K15; 35Q53. Secondary 35Q15; 34A55

Type
Research Article
Copyright
2004 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)