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KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions

A. Sakhnovich — 2012

Mathematical Modelling of Natural Phenomena

The matrix KdV equation with a negative dispersion term is considered in the right upper quarter–plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial–boundary conditions.

On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

A. Sakhnovich — 2010

Mathematical Modelling of Natural Phenomena

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the -wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac...

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