Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 78-125 (2005).
Summary: This is a review of the main ideas of the inverse scattering method for solving nonlinear evolution equations, known as soliton equations. As a basic tool we use the fundamental analytic solutions of the Lax operator . Then the inverse scattering problem for reduces to a Riemann-Hilbert problem. Such construction has been applied to a wide class of Lax operators, related to the simple Lie algebras. We construct the kernel of the resolvent of in terms of and derive the spectral decompositions of . Thus we can solve the NLS equation and its multi-component generalizations, the -wave equations, etc. Applying the dressing method of Zakharov and Shabat we derive the -soliton solutions of these equations.
|35Q55||NLS-like (nonlinear Schrödinger) equations|
|37K15||Integration of completely integrable systems by inverse spectral and scattering methods|