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Towards an Inverse Scattering theory for non decaying potentials of the heat equation

M Boiti·F Pempinelli·A K Pogrebkov·B Prinari
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Abstract

The resolvent approach is applied to the spectral analysis of the heat equation with non decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe $N$ solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a DBAR-problem explicitly in terms of the corresponding objects associated to the original potential. Regularity conditions of the potential in the cases N=1 and N=2 are investigated in details. The singularities of the resolvent for the case N=1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.

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M Boiti, F Pempinelli, A K Pogrebkov, B Prinari. 2001-01-17. Towards an Inverse Scattering theory for non decaying potentials of the heat equation. https://doi.org/10.1088/0266-5611%2F17%2F4%2F324

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