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On the Moutard transformation for integrable partial differential equations

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.