Source and sink solution, finite time blow-up, diffusion, creation and annihilation processes in the Davey–Stewartson equation

Abstract

Making use of the theory of symmetry transformations in PDE we construct new solutions of the Davey–Stewartson (DS) equation. First, among its reductions one can find a time independent like-DS equation. Starting with the expressions of the well-known coherent structures of the DS equation, we can obtain solutions of this time independent equation. From these solutions, families of solutions of DS equation depending on three arbitrary functions on t are obtained. Besides, new solutions can also be constructed by applying some elements of the symmetry group to known solutions of the model. Among the solutions constructed using both approaches, one can find source and sink solutions, solutions describing the creation, the diffusion or annihilation of a dromion (or in general, a set of localized structures), finite time blow-up processes, instantaneous source solutions, and coherent structures moving at arbitrary velocities along arbitrary curves.