Inverse Spectral Problem for the Periodic Camassa-Holm Equation

DOI

10.2991/jnmp.2004.11.4.6How to use a DOI?

Abstract

We consider the direct/inverse spectral problem for the periodic Camassa-Holm eqution. In fact, we survey the direct/inverse spectral problem for the periodic weighted operator Ly = m-1 (-y +1 4 y) acting in the space L2 (R, m(x)dx), where m = uxx-u > 0 is a 1-periodic positive function and u is the solution of the Camassa-Holm equation ut - uxxt + 3uux = 2uxuxx + uuxxx. For the operator L we describe the complete solution of the inverse spectral problem: i) uniqueness, prove that the spectral data uniquely determines the potential, ii) characterization, give conditions for some data to be the spectral data of some potential, iii) reconstruction, give an algorithm for recovering the potential from the spectral data, iv) a priori estimates, obtain two-sided a priori estimates of u, m in terms of gap lengths.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Evgeni Korotyaev
PY  - 2004
DA  - 2004/11/01
TI  - Inverse Spectral Problem for the Periodic Camassa-Holm Equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 499
EP  - 507
VL  - 11
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2004.11.4.6
DO  - 10.2991/jnmp.2004.11.4.6
ID  - Korotyaev2004
ER  -