Davey-Stewartson type equations in 4+2 and 3+1 possessing soliton solutions

An integrable generalisation of a Davey-Stewartson type system of two equations involving two scalar functions in 4+2, i.e., in four spatial and two temporal dimensions, has been recently derived by one of the authors. Here, we first show that there exists a reduction of this system to a single equation involving a scalar function in 4+2; we will refer to this equation as the 4+2 Davey-Stewartson equation. We then show that it is possible to reduce this equation to an equation in 3+1, which we will refer to as the 3+1 Davey-Stewartson equation. Furthermore, by employing the so-called direct linearising method, we compute 1- and 2-soliton solutions for both the 4+2 and the 3+1 Davey-Stewartson equations.