Inverse problems and nonlinear evolution equations
Solutions, Darboux Matrices and Weyl–Titchmarsh Functions
This book explores the method of operator identities and the theory of S-nodes, as developed by Lev Sakhnovich, emphasizing the crucial role of the transfer matrix function generated by the S-node. The authors present fundamental solutions to significant systems of differential equations through this transfer matrix function, either directly or via representations involving the corresponding Darboux matrix, particularly when considering Bäcklund–Darboux transformations and explicit solutions. The transfer matrix function also facilitates solving an inverse problem, specifically recovering a system from its Weyl function. The work covers Weyl theories of selfadjoint and skew-selfadjoint Dirac systems, related canonical systems, discrete Dirac systems, and systems associated with the N-wave equation, among others. The findings on direct and inverse problems are applied to investigate initial-boundary value problems for integrable (nonlinear) wave equations using the inverse spectral transformation method. The evolution of the Weyl function and solutions for initial-boundary value problems in a semi-strip are derived for various important nonlinear equations. Additionally, the book details some uniqueness and global existence results using evolution formulas. A basic understanding of linear algebra, calculus, and operator theory from standard university courses is sufficient for readers.
