Bäcklund-Darboux Transformations and Melnikov Analysis for Davey-Stewartson II Equations

The Bäcklund-Darboux transformations are successfully utilized to construct heteroclinic orbits of Davey-Stewartson II equations through an elegant iteration of the transformations. In [17], we successfully built Melnikov vectors with the gradients of Floquet discriminants. Since there is no Floquet discriminant for Davey-Stewartson equations (in contrast to nonlinear Schrödinger equations [17]), the Melnikov vectors here are built with the novel idea of replacing the gradients of Floquet discriminants by quadratic products of Bloch functions. Such Melnikov vectors still maintain the properties of Poisson commuting with the gradient of the Hamiltonian and exponential decay as time approaches positive and negative infinities. This solves the problem of building Melnikov vectors for Davey-Stewartson equations without using the gradients of a Floquet discriminant. Melnikov functions for perturbed Davey-Stewartson II equations evaluated on the above heteroclinic orbits are built.