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How many types of soliton solutions do we know? (English) Zbl 1101.35069
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 7th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 2–10, 2005. Sofia: Bulgarian Academy of Sciences (ISBN 954-8495-30-9/pbk). 11-34 (2006).
Summary: We discuss several ways of how one could classify the various types of soliton solutions related to nonlinear evolution equations that are solvable with the generalized $n×n$ Zakharov-Shabat system. In doing so we make use of the fundamental analytic solutions, the dressing procedure and other tools characteristic for the inverse scattering method. We propose to relate to each subalgebra $\mathrm{𝔰𝔩}\left(p\right)$, $2\le p\le n$ of $\mathrm{𝔰𝔩}\left(n\right)$, a type of one-soliton solutions which have $p-1$ internal degrees of freedom.
MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 37K30 Relations of infinite-dimensional systems with algebraic structures 37K15 Integration of completely integrable systems by inverse spectral and scattering methods 35Q51 Soliton-like equations