Davey Stewartson Research Articles

Symmetry Analysis and Similarity Solutions of a Resonant Davey–Stewartson System

Lie point symmetry analysis is performed to a recently proposed (2+1)-dimensional nonlinear system, the resonant Davey–Stewartson equation. Some similarity solutions of the RDS equation are thus obtained.

A note on the Cauchy problem of the generalized Davey–Stewartson equations

In this note, we establish some local and global existence results for the Cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear Schrödinger equations and the Davey–Stewartson equations. These results improve some previously obtained results by some other authors when they are restricted to certain special equations.

Self-similar solutions to a generalized Davey–Stewartson system

In this paper we study the global solvability and existence of self-similar solutions to a generalized Davey–Stewartson system. We first use the Banach fixed point theorem to establish a general global existence theorem. By using this general result to a class of special initial data we obtain the existence of global self-similar solutions.

The Davey–Stewartson equation with sources derived from nonlinear variable separation method

In this paper, by taking the Davey–Stewartson equation as an example, a new construction procedure based on nonlinear variable separation method is presented to obtain nonlinear evolution equations with sources for the first time.

On the relationship between nonlinear equations integrable by the method of characteristics and equations associated with commuting vector fields

It was shown recently that Frobenius reduction of the matrix fields reveals interesting relations among the nonlinear partial differential equations (PDEs) integrable by the inverse spectral transform method (S-integrable PDEs), linearizable by the Hopf–Cole substitution (C-integrable PDEs), and integrable by the method of characteristics (Ch-integrable PDEs). However, only two classes of S-integrable PDEs have been involved: soliton equations (such as Korteweg–de Vries, nonlinear Schrödinger, Kadomtsev–Petviashvili, and Davey–Stewartson equations) and GL(N,C) self-dual-type PDEs (such as Yang–Mills equation). In this paper we consider the simple five-dimensional nonlinear PDE from another class of S-integrable PDEs, namely, scalar nonlinear PDE which is the commutativity condition of the pair of vector fields. We show its origin from the (1+1)-dimensional hierarchy of Ch-integrable PDEs after certain composition of Frobenius type and differential reductions imposed on the matrix fields. Matrix generalization of the above scalar nonlinear PDE will be derived as well.

A note on the global existence of small amplitude solutions to a generalized Davey–Stewartson system

In this paper, we are interested in the Cauchy problem for a generalized Davey–Stewartson (GDS) system. We establish the global time existence of small mass solutions for the GDS system in the elliptic–hyperbolic–hyperbolic case.

Dromion solutions of noncommutative Davey–Stewartson equations

We consider a noncommutative version of the Davey–Stewartson equations and derive two families of quasideterminant solution via Darboux and binary Darboux transformations. These solutions can be verified by direct substitution. We then calculate the dromion solutions of the equations and obtain computer plots in a noncommutative setting.

Propagating wave patterns for the ‘resonant’ Davey–Stewartson system

The resonant nonlinear Schrödinger (RNLS) equation exhibits the usual cubic nonlinearity present in the classical nonlinear Schrödinger (NLS) equation together with an additional nonlinear term involving the modulus of the wave envelope. It arises in the context of the propagation of long magneto-acoustic waves in cold, collisionless plasma and in capillarity theory. Here, a natural (2 + 1) (2 spatial and 1 temporal)-dimensional version of the RNLS equation is introduced, termed the ‘resonant’ Davey–Stewartson system. The multi-linear variable separation approach is used to generate a class of exact solutions, which will describe propagating, doubly periodic wave patterns.

The Davey-Stewartson Equation on the Half-Plane

The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrodinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI.

Finding general and explicit solutions of high nonlinear equations by the Exp-Function method

In this paper, Davey–Stewartson equation and coupled Klein–Gordon–Schrödinger (KGS) equations and homogeneous nonlinear convection–diffusion problem are solved using the Exp-function method. The capabilities and wide-range applications of the Exp-function method are illustrated. This method can be used as an alternative to obtain analytical and approximate solutions of different types of differential equations in engineering mathematics.

Modulational instability of dust acoustic waves in dusty plasmas with ultraviolet irradiation and nonadiabatic dust charge variation

The combined effects of nonadiabatic dust charge fluctuation, ultraviolet irradiation, and transverse perturbation on the modulational instability of nonlinear dust acoustic (DA) waves in dusty plasma consisting of Boltzmann-distributed electrons, ions, as well as positively charged dust grain were presented here. By using the reductive perturbation technique, a three-dimensional modified Davey–Stewartson equation governing the nonlinear waves is derived. It is shown that the modulation properties of the DA waves were strongly affected by the nonadiabatic dust charge variation and photoelectrons and there was a maximum time for the modulational instability.

Two-dimensional envelope electron-acoustic waves under transverse perturbations

As is well known, the envelope electron-acoustic (EA) nonlinear waves in one dimension are governed by the nonlinear Schrödinger equation. If transverse perturbations are considered, then this type of nonlinear wave can be described by the general form of the Davey–Stewartson equation. In this work, modulational properties of EA wave and its stability regions in two-dimensional plasma have been studied.

On the existence of special solutions of the generalized Davey–Stewartson system

In this note we show that for certain choice of parameters the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system admits time-dependent travelling wave solutions of the kind given in [V.A. Arkadiev, A.K. Pogrebkov, M.C. Polivanov, Inverse scattering transform method and soliton solutions for Davey–Stewartson II equation, Physica D 36 (1989) 189–197] for the hyperbolic Davey–Stewartson system. These solutions lead to radial solutions as well. We also find the sufficient conditions for non-existence of travelling wave solutions for the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system by using the point of view developed in [A. Eden, T.B. Gürel, E. Kuz, Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system, IMA J. Appl. Math. (in press)].

A cross-constrained variational problem for the generalized Davey–Stewartson system

AbstractWe study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson systemin ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering

We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(|x|^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$-solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type.We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.

The relationship between the hyperbolic Nizhnik–Novikov–Veselov equation and the stationary Davey–Stewartson II equation

A Lax system in three variables is presented, two equations of which form the Lax pair of the stationary Davey–Stewartson II equation. With certain nonlinear constraints, the full integrability condition of this Lax system contains the hyperbolic Nizhnik–Novikov–Veselov equation and its standard Lax pair. The Darboux transformation for the Davey–Stewartson II equation is used to solve the hyperbolic Nizhnik–Novikov–Veselov equation, and global multi-soliton solutions are obtained. It is proved that the n × n soliton solution derived by the Darboux transformation of order n approaches zero uniformly and exponentially at spatial infinity and is asymptotic to n2 solitons at temporal infinity.

Wind-forced modulations of finite-depth gravity waves

It has been shown recently that the modulations of deep-water gravity waves grow superexponentially under the effect of wind [S. Leblanc, “Amplification of nonlinear surface waves by wind,” Phys. Fluids 19, 101705 (2007)]. Finite depth is considered here. Based on Davey–Stewartson equations with forcing, stability analysis shows that longitudinal modulations are amplified when kh>1.363, where k is the wave number of the carrier wave and h is the water depth, but that oblique modulations dominate when 0<kh<2.199. In each case, amplification is characterized in the linear stage by superexponential growth. This latter is, however, so small when kh<0.380 that no modulation is expected in shallow water, although the carrier wave still grows, thanks to wind forcing.

Symmetry, full symmetry groups, and some exact solutions to a generalized Davey–Stewartson system

The Lie symmetry algebra of a generalized Davey–Stewartson (GDS) system is obtained. The general element of this algebra depends on eight arbitrary functions of time, which has a Kac–Moody–Virasoro loop algebra structure and is isomorphic to that of the standard integrable Davey–Stewartson equations under certain conditions imposed on parameters and arbitrary functions. Then based on the symmetry group direct method proposed by Lou and Ma [J. Phys. A 38, L129 (2005)] the full symmetry groups of the GDS system are obtained. From the full symmetry groups, both the Lie symmetry group and a group of discrete transformations can be obtained. Finally, some exact solutions involving sech-sech2-sech2 and tanh-tanh2-tanh2 type solitary wave solutions are presented by a generalized subequation expansion method.

Soliton and chaotic structures of dust ion-acoustic waves in quantum dusty plasmas

The quantum hydrodynamic model is employed to study the soliton and chaotic structures of dust ion-acoustic waves in quantum dusty plasmas consisting of electrons, ions, and negatively/positively charged dust particles. By means of the reductive perturbation technique, two-dimensional Davey–Stewartson (DS) system is derived. By improving the extended projective method and the extended tanh-function method, a separation of variables solution with arbitrary functions for the Davey–Stewartson system is obtained. Many soliton and chaotic structures such as localized nonlinear coherent structure, line-soliton structure, periodic wave pattern structure, Rössler and Lorenz chaotic structures are given. It is found that these structures are effected by the quantum effects.

(2+1)-dimensional Davey–Stewartson I description in a 2D lattice model

The paper is devoted to investigate a two-dimensional (2D) lattice model with interactions between the first- and second-nearest neighbors, which is extracted from a cubic lattice of cystalline alloys. With the help of the method of multiple scales combined with a quasidiscreteness approximation, the (2+1)-dimensional Davey–Stewartson-I (DS-I) equation is obtained from the original 2D discrete system, which is completely integrable and admits explicit solutions such as positon solutions (which are locally singular) and dromions (which decay exponentially in all spatial directions). Then the leading order-approximated solutions of the 2D lattice are constructed by the explicit solutions of the DS-I.