Misc,
Basic aspects of soliton theory
(2006)cite arxiv:nlin/0604004Comment: Reported at the Sixth International Conference on Geometry, Integrability and Quantization, June 3-10, 2004, Varna, Bulgaria. Published in Ivailo M. Mladenov and Allen C. Hirshfeld, Editors, SOFTEX, Sofia 2005.
Abstract
This is a review of the main ideas of the inverse scattering method (ISM) for
solving nonlinear evolution equations (NLEE), known as soliton equations. As a
basic tool we use the fundamental analytic solutions (FAS) of the Lax operator
L. Then the inverse scattering problem for L reduces to a Riemann-Hilbert
problem. Such construction cab be applied to wide class of Lax operators,
related to the simple Lie algebras g. We construct the kernel of the resolvent
of L in terms of FAS and derive the spectral decompositions of L. Thus we can
solve the relevant classes of NLEE which include the NLS eq. and its
multi-component generalizations, the N-wave equations etc. Using the dressing
method of Zakharov and Shabat we derive the N-soliton solutions of these
equations. We explain that the ISM is a natural generalization of the Fourier
transform method. As appropriate generalizations of the usual exponential
function we use the so-called "squared solutions" which are constructed again
in terms of FAS and the Cartan-Weyl basis of the relevant Lie algebra. One can
prove the completeness relations for the "squared solutions" which in fact
provide the spectral decompositions of the recursion operator Łambda. These
decompositions can be used to derive all fundamental properties of the
corresponding NLEE in terms of Łambda: i) the explicit form of the class of
integrable NLEE; ii) the generating functionals of integrals of motion; iii)
the hierarchies of Hamiltonian structures. We outline the importance of the
classical R-matrices for extracting the involutive integrals of motion.
