The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach

被引：0
作者：
Anne Boutet de Monvel
Dmitry Shepelsky
机构：
[1] Université Paris Diderot Paris 7,Institut de Mathematiques de Jussieu
[2] Verkin Institute for Low Temperature Physics,Mathematical Division
来源：
Journal of Geometric Analysis | 2008年 / 18卷
关键词：
Inverse scattering; Riemann–Hilbert; Initial-boundary value problem; Camassa–Holm; 37K15; 35Q53; 30E25; 37K40;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation ut−utxx+2ux+3uux=2uxuxx+uuxxx on the half-line x≥0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann–Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation.
引用
收藏
页码：285 / 323
页数：38
相关论文
共 41 条
[1]  
Boutet de Monvel A.(2003)Analysis of the global relation for the nonlinear Schrödinger equation on the half-line Lett. Math. Phys. 65 199-212
[2]  
Fokas A.S.(2004)The mKdV equation on the half-line J. Inst. Math. Jussieu 3 139-164
[3]  
Shepelsky D.(2004)Initial boundary value problem for the mKdV equation on a finite interval Ann. Inst. Fourier (Grenoble) 54 1477-1495
[4]  
Boutet de Monvel A.(2006)Riemann–Hilbert approach for the Camassa–Holm equation on the line C. R. Math. Acad. Sci. Paris 343 627-632
[5]  
Fokas A.S.(1993)An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1661-1664
[6]  
Shepelsky D.(1994)A new integrable shallow water equation Adv. Appl. Mech. 31 1-33
[7]  
Boutet de Monvel A.(2001)On the scattering problem for the Camassa–Holm equation R. Soc. Lond. Proc. Ser. A: Math. Phys. Eng. Sci. 457 953-970
[8]  
Shepelsky D.(2006)Inverse scattering transform for the Camassa–Holm equation Inverse Prob. 22 2197-2207
[9]  
Boutet de Monvel A.(2003)On the inverse scattering approach to the Camassa–Holm equation J. Nonlinear Math. Phys. 10 252-255
[10]  
Shepelsky D.(1999)A shallow water equation on the circle Commun. Pure Appl. Math. 52 949-982
12345