A generalisation in 2+1 dimensions of the Korteweg-de Vries equation is related to the spectral problem ( delta x2- delta y2-p(x,y)) phi (x,y;k)=0. It can contain arbitrary functions of x+y or x-y and time. The Cauchy problem, associated with initial data decaying sufficiently rapidly at infinity, is linearised by an extension of the spectral transform technique to two spatial dimensions. The spectral data are explicitly defined in terms of the initial data and the inverse problem is formulated as a non-local Riemann-Hilbert boundary-value problem. The presence of arbitrary functions of x+y and x-y in the evolution equation implies that the time evolution of the spectral data is linear but non-local. Discrete spectral data are forbidden and, consequently, localised soliton solutions are not allowed.

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A two-dimensional nonlinear evolution equation is solved in the inverse spectral transform scheme. It coincides, when reduced to one spatial dimension, with the dispersive long wave equation. The Backlund transformation, soliton solution and superposition formula are obtained. The spectral transform is explicitly defined and the corresponding linear evolution of the spectral data is given. The inverse spectral problem is formulated as a non-local Riemann-Hilbert boundary value problem and solved.

A two-dimensional generalisation of the sine- and the sinh-Gordon equations, which one refers to as the shine-Gordon equations, is obtained and solved through the inverse spectral transform (IST) method. The Backlund transformation and nonlinear superposition formula are constructed and explicit wave solitons are given. It is shown also that a slightly different procedure furnishes an IST-solvable extension in 2+1 dimensions of the dispersive long-wave equation.

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.