6533b873fe1ef96bd12d554f

RESEARCH PRODUCT

Trivial S-Matrices, Wigner-Von Neumann Resonances and Positon Solutions of the Integrable Nonlinear Evolution Equations

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It is well known that the scattering matrix is different from the unit matrix in the case of 1-dimensional Schrodinger operator with smooth rapidly decreasing nonzero potential. This no more true in the case of the slowly decreasing and oscillating potentials for which the absence of scattering is accompanied by the occurrence of the Wigner-von Neumann resonances embedded in the positive absolutely continuous spectrum. Taken as initial conditions in the KdV like integrable partial differential equations these potentials generate interesting family of explicit solutions. Below we will call them positon or multipositon solutions. The interaction of an arbitrary finite number of positons and solitons can also be introduced in a natural way in terms of wronskian determinants. The aim of this talk is to describe the main properties of the multipositon-soliton solutions for the KdV model. In particular it is shown that positons survive a mutual collision unchanged. For an interaction between positons and solitons it is demonstrated that the soliton is unchanged by the collision. The positon, by contrast, acquires two additional but always finite phase shifts.