The Riemann-Hilbert Formalism For Certain Linear and Nonlinear Integrable PDEs

DOI

10.2991/jnmp.2007.14.3.12How to use a DOI?

Abstract

We show that by deforming the Riemann-Hilbert (RH) formalism associated with certain linear PDEs and using the so-called dressing method, it is possible to derive in an algorithmic way nonlinear integrable versions of these equations. In the usual Dressing Method, one first postulates a matrix RH problem and then constructs dressing operators. Here we present an algorithmic construction of matrix Riemann-Hilbert (RH) problems appropriate for the dressing method as opposed to postulating them ad hoc. Furthermore, we introduce two mechanisms for the con- struction of the relevant dressing operators: The first uses operators with the same dispersive part, but with different decay at infinity, while the second uses pairs of operators corresponding to different Lax pairs of the same linear equation. As an application of our approach, we derive the NLS, derivative NLS, KdV, modified KdV and sine-Gordon equations.

Copyright

© 2007, 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  - Dimitrios A. Pinotsis
PY  - 2007
DA  - 2007/10/01
TI  - The Riemann-Hilbert Formalism For Certain Linear and Nonlinear Integrable PDEs
JO  - Journal of Nonlinear Mathematical Physics
SP  - 474
EP  - 493
VL  - 14
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2007.14.3.12
DO  - 10.2991/jnmp.2007.14.3.12
ID  - Pinotsis2007
ER  -