Basic aspects of soliton theory. (English) Zbl 1086.35102

Gerdjikov, Vladimir S.

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 equations (nonlinear Schrödinger equations)
37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51	Soliton equations
35Q15	Riemann-Hilbert problems in context of PDEs

Keywords:

review; Lax operator; simple Lie algebras; resolvent; NLS equation; \(N\)-wave equations; \(N\)-soliton solutions