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Zbl 1086.35102
Gerdjikov, Vladimir S.
Basic aspects of soliton theory. (English)
[A] Mladenov, Iva\"ilo 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. 78-125 (2005). ISBN 954-84952-9-5/pbk

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.

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
37K15 Integration by inverse spectral and scattering methods
35Q51 Solitons
35Q15 Riemann-Hilbert problems

Keywords: review; Lax operator; simple Lie algebras; resolvent; NLS equation; $N$-wave equations; $N$-soliton solutions

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