Discrete and Continuous Nonlinear Schrödinger Systems
Cambridge University Press, 2004 - Mathematics - 257 pages
During the past 30 years there have been important and far reaching developments in the study of nonlinear waves including "soliton equations", a class of nonlinear wave equations which arise frequently in applications. The wide interest in this field can be traced to the understanding of certain special stable, localized waves called solitons and the associated development of a method of solution to a class of nonlinear wave equations. Prior to these developments very little was known about the solutions to these equations.The method of solution, termed the inverse scattering transform (IST), applies to a class of continuous and discrete nonlinear Schrödinger (NLS) equations. NLS equations are of particular interest because they arise in many important physical applications, such as nonlinear optics, fluid dynamics and statistical physics. Many of the details of the IST presented in this book are not available in the previously-published literature. This book provides a thorough, self-contained presentation of the IST as applied to NLS systems.
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algebraic analytic functions associated with zj assume asymptotic boundary conditions compute conserved quantities continuous convergent corresponding defined denote derived deta(z difference equation discrete eigenvalues eigenfunctions eigenvalue zj eigenvalues fiber finite follows formula Gel'fand-Levitan-Marchenko given Green's functions Hamiltonian Hence IDNLS integrable discrete integral equations inverse problem inverse scattering transform Jost functions lattice Lax pair Lemma linear matrix with index meromorphic Mn(z Moreover Neumann series NLS equation NLS systems nonlinear evolution equation nonlinear ladder network norming constant associated Note number of eigenvalues obtain one-soliton solution optical fibers pair pairwise poles potentials Q proper eigenvalues reconstruct reflection coefficients region relation respectively Riemann-Hilbert problem satisfy the symmetry scattering coefficients scattering data scattering problem soliton interactions soliton solutions solve spectral parameter substituting summation equation Symmetry 5.3 taking into account theory time-dependence tions Toda lattice unit circle VNLS Wronskian yields zero of a(z