The Camassa–Holm Equation on the Half-Line: a Riemann–Hilbert Approach |
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Authors: | Anne Boutet de Monvel Dmitry Shepelsky |
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Institution: | (1) Institut de Mathematiques de Jussieu, Université Paris Diderot Paris 7, 175 rue du Chevaleret, 75013 Paris, France;(2) Mathematical Division, Verkin Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine |
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Abstract: | We consider the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation u
t
−u
txx
+2u
x
+3uu
x
=2u
x
u
xx
+uu
xxx
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.
Dedicated to Gennadi Henkin in great admiration. |
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Keywords: | Inverse scattering Riemann– Hilbert Initial-boundary value problem Camassa– Holm |
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