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Basic aspects of soliton theory. (English) Zbl 1086.35102
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 $\chi^\pm(x,\lambda)$ of the Lax operator $L(\lambda)$. Then the inverse scattering problem for $L(\lambda)$ 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 $L(\lambda)$ in terms of $\chi^\pm(x,\lambda)$ and derive the spectral decompositions of $L(\lambda)$. Thus we can solve the NLS equation and its multi-component generalizations, the $N$-wave equations, etc. Applying the dressing method of Zakharov and Shabat we derive the $N$-soliton solutions of these equations. For the entire collection see [Zbl 1066.53003].

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 37K15 Integration of completely integrable systems by inverse spectral and scattering methods 35Q51 Soliton-like equations 35Q15 Riemann-Hilbert problems