No abstract

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 Bäcklund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a ∂-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.

We study the initial value problem of the Kadomtsev-Petviashvili I (KPI) equation with initial data u(x 1 , x 2 , 0) = u 1 (x 1 )+u 2 (x 1 , x 2 ), where u 1 (x 1 ) is the one-soliton solution of the Korteweg-de Vries equation evaluated at zero time and u 2 (x 1 , x 2 ) decays sufficiently rapidly on the (x 1 , x 2 )-plane. This involves the analysis of the nonstationary Schrödinger equation (with time replaced by x 2 ) with potential u(x 1 , x 2 , 0). We introduce an appropriate sectionally analytic eigenfunction in the complex k-plane where k is the spectral parameter. This eigenfunction has the novelty that in addition to the usual jump across the real k-axis, it also has a jump across a segment of the imaginary k-axis. We show that this eigenfunction can be reconstructed through a linear integral equation uniquely defined in terms of appropriate scattering data. In turn, these scattering data are uniquely constructed in terms of u 1 (x 1 ) and u 2 (x 1 , x 2 ). This result implies that the solution of the KPI equation can be obtained through the above linear integral equation where the scattering data have a simple t-dependence.

The inverse scattering theory of the heat equation is developed for a special subclass of potentials nondecaying at space infinity-perturbations of the one-soliton potential by means of decaying two-dimensional functions. Extended resolvent, Green's functions, and Jost solutions are introduced and their properties are investigated in detail. The singularity structure of the spectral data is given and then the inverse problem is formulated in an exact distributional sense.

Lattice soliton: Solutions to the Kadomtsev-Petviashvili equation with positive dispersion