Many equations can be expressed as a cubic nonlinear Schrödinger (NLS) equation with additional terms, such as the Davey-Stewartson (DS) system [1]. As it is the case for the NLS equation, the solutions of the DS system are invariant under the pseudo-conformal transformation. For the elliptic NLS, this invariance plays a key role in understanding the blow-up profile of solutions, whereas in the...

We have used the homotopy analysis method (HAM) to obtain solution of Davey-Stewartson equations of fractional order. The fractional derivative is described in the Caputo sense. The results obtained by this method have been compared with the exact solutions. Stability and convergence of the proposed approach is investigated. The effects of fractional derivatives for the systems under considerat...

The Davey-Stewartson equations are used to describe the long time evolution of a threedimensional packets of surface waves. Assuming that the argument functions are quadratic in spacial variables, we find in this paper various exact solutions modulo the most known symmetry transformations for the Davey-Stewartson equations.

We have recently shown [1] that fully-localized threedimensional wave envelopes (so-called dromions) can exist and propagate on the surface of ice-covered waters. Here we show that the inertia of the ice can play an important role in the size, direction and speed of propagation of these structures. We use multiple-scale perturbation technique to derive governing equations for the weakly nonline...

The Davey-Stewartson 1(DS1) system[9] is an integrable model in two dimensions. A quantum DS1 system with 2 colour-components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 colour-components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in o...

The Davey-Stewartson 1(DS1) system[9] is an integrable model in two dimensions. A quantum DS1 system with 2 colour-components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 colour-components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in o...

Purely dispersive partial differential equations such as the Korteweg–de Vries equation, the nonlinear Schrödinger equation, and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in...

In this paper, we consider applications of the connection between soliton theory and commuting nonselfadjoint operator theory, established by Livšic Avishai. An approach to inverse scattering problem wave equations is presented, based on colligation (or vessel theory) in case bounded operators a Hilbert space, when one belongs larger class nondissipative with asymptotics corresponding curves. T...

The exact solutions to the Davey-Stewartson 1 equations are constructed to describe the development of a transversal solitary wave instability. Two types of plane dark soliton decays into two-dimensional solitary waves are found and analysed. It is also shown that the transversal instability of one-dimensional waveguides leads only to a shift of their drift velocity under the action of nonlocal...