The Moutard transformation is a Darboux-type transformation appropriate to linear scattering problems of hyperbolic or elliptic type in two independent and one dependent variable. The authors derive this transformation, as well as the Darboux transformation for parabolic linear problems, by a factorization method and discuss the use of compositions of such transformations in the construction of solutions to the Novikov-Veselov equations.
Generalizations of the N-wave, Oavey-Stewartson, and Kadomtsev-Petviashvili equations associated with homogeneous and symmetric spaces are presented. These equations are (2 + 1 )-dimensional generalizations of those presented by Fordy and Kulish [Commun.Math. Phys. 89, 427 (1983) 1 and Athorne and Fordy [J. Phys. A 20,1377Phys. A 20, (1987]. Examples are explicitly presented that are associated with the simplest spaces. In particular, a single component, (2 + 1) -dimensional generalization of the KdV equation is presented.2018
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