Davey Stewartson Research Articles

Homoclinic orbits of the Davey-Stewartson equations

The homoclinic solutions problem of the Davey-Stewartson (DS) Equations were studied. By using the Hirota’s bilinear method, the homoclinic orbits of the Davey-Stewartson Equations were obtained through the dependent variable transformation. The homoclinic orbits of the Davey-Stewartson Equations were discussed.

Global existence and nonexistence results for a generalized Davey–Stewartson system

We consider a system of three equations, which will be called generalized Davey–Stewartson equations, involving three coupled equations, two for the long waves and one for the short wave propagating in an infinite elastic medium. We classify the system according to the signs of the parameters. Conserved quantities related to mass, momentum and energy are derived as well as a specific instance of the so-called virial theorem. Using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic–elliptic–elliptic case.

Modulational instability of multi-dimensional dust ion-acoustic waves

The modulation of dust-ion acoustic wave (DIAW) in unmagnetized dusty plasmas with the transverse plane perturbations is studied. By using the standard reductive perturbation method, a three-dimensional Davey–Stewartson (3D DS) equation in unmagnetized dusty plasmas is obtained. The modulational instability of DIAW described by the 3D DS equation is investigated. Some new and important explicit criterion for the modulational instability of DIAW described by 3D DS equation is obtained. It is shown that the modulation properties of DIAW in 3D dusty plasmas are very different from that in 1D case.

Projective differential geometry of higher reductions of the two-dimensional Dirac equation

We investigate reductions of the two-dimensional Dirac equation imposed by the requirement of the existence of a differential operator D n of order n mapping its eigenfunctions to adjoint eigenfunctions. For first order operators these reductions (and multicomponent analogs thereof) lead to the Lame equations descriptive of orthogonal coordinate systems. Our main observation is that nth order reductions coincide with the projective-geometric ‘Gauss–Codazzi’ equations governing special classes of line congruences in the projective space P 2 n−1 , which is the projectivised kernel of D n . In the second order case this leads to the theory of W-congruences in P 3 which belong to a linear complex, while the third order case corresponds to isotropic congruences in P 5. Higher reductions are compatible with odd order flows of the Davey–Stewartson hierarchy. All these flows preserve the kernel D n , thus defining nontrivial geometric evolutions of line congruences. Multicomponent generalizations are also discussed. The correspondence between geometric picture and the theory of integrable systems is established; the definition of the class of reductions and all geometric objects in terms of the multicomponent KP hierarchy is presented. Generating forms for reductions of arbitrary order are constructed.

The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations

The periodic wave solutions for a class of nonlinear partial differential equations, including the Davey–Stewartson equations and the generalized Zakharov equations, are obtained by using the F-expansion method, which can be regarded as an overall generalization of the Jacobi elliptic function expansion method recently proposed. In the limit cases the solitary wave solutions of the equations are also obtained.

(2+1) Dimensional Wave Patterns of the Davey–Stewartson System

(2+1) (2 spatial and 1 temporal) dimensional patterns of standing waves are calculated theoretically for the Davey–Stewartson equations (DS). DS are relevant in hydrodynamic free surface waves of f...

Two-dimensional envelop ion acoustic wave under transverse perturbations

As is well known, the envelop ion acoustic nonlinear waves are expressed by the Nonlinear Schrodinger Equation (NLSE). In this paper, we find that under the transverse perturbations, this kind of nonlinear waves can be described by Davey–Stewartson (DS) equation. One of the specific two-dimensional envelop soliton is given for the general DS equation and its stability regions have been investigated.

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

SYMBOLIC COMPUTATION AND WEIERSTRASS ELLIPTIC FUNCTION SOLUTIONS OF THE DAVEY–STEWARTSON (DS) SYSTEM

In this paper, with the symbolic computation, the doubly-periodic solutions of the Davey–Stewartson (DS) system, which describes the modulational instability of uniform train of weakly nonlinear water waves in the two-dimensional space, are investigated in terms of the Weierstrass elliptic function ℘(ξ; g2, g3). In particular, we also give new solitary wave solutions of this system.

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

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.

The Davey–Stewartson I equation on the quarter plane with homogeneous Dirichlet boundary conditions

Dromions are exponentially localized coherent structures supported by nonlinear integrable evolution equations in two spatial dimensions. In the study of initial-value problems on the plane, such solutions occur only if one imposes nontrivial boundary conditions at infinity, a situation of dubious physical significance. However, it is established here that dromions appear naturally in the study of boundary-value problems. In particular, it is shown that the long time asymptotics of the solution of the Davey–Stewartson I equation in the quarter plane with arbitrary initial conditions and with zero Dirichlet boundary conditions is dominated by dromions. The case of nonzero Dirichlet boundary conditions is also discussed.

The n-component KP hierarchy and representation theory

It is the aim of the present article to give all formulations of the n-component KP hierarchy and clarify connections between them. The generalization to the n-component KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey–Stewartson system (for n=2), the two-dimensional Toda lattice (for n=2), the n-wave system (for n⩾3) and the Darboux–Egoroff system. It also allows us to construct natural generalizations to the Davey–Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the n-component KP hierarchy allows us to construct their solutions by making use of vertex operators.

Weakly two-dimensional modulated wave packet in dusty plasmas

The higher order transverse perturbations for modulated wave packets in a dusty plasma are investigated. A Davey–Stewartson equation is obtained for this system. The instability for small amplitude linear transverse perturbations are investigated.

Two-dimensional wave packets in an elastic solid with couple stresses

The problem of (2+1) (two spatial and one temporal) dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. The aim is to derive (2+1) non-linear model equations for the description of elastic waves in the far field. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that the modulation of waves is governed by a system of three non-linear evolution equations. These equations involve amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave, and will be called the “generalized Davey–Stewartson equations”. Under some restrictions on parameter values, the generalized Davey–Stewartson equations reduce to the Davey–Stewartson and to the non-linear Schrödinger equations. Finally, some special solutions involving sech–tanh–tanh and tanh–tanh–tanh type solitary wave solutions are presented.

Instability and collapse of waveguides on the water surface under the ice cover

We consider long gravity-flexural waves on a surface of a perfect liquid of a finite depth under elastic ice-plate. Such waves of small but finite amplitude are governed by the generalized Kadomtsev–Petviashvili (KP) equation, which contains higher spatial derivatives. The generalized KP equation admits waveguide solutions, describing waves periodic in the direction of propagation and localized in the transverse direction. Waveguide represents a wave being the nonlinear product of instability of carrier monochromatic wave with respect to transverse perturbations. Instability of waveguides in its nonlinear stage is studied. For this purpose we use the alternative description via the Davey–Stewartson (DS) equations for slowly varying amplitudes of monochromatic waves. The DS equations are asymptotically equivalent to the initial generalized KP equation. Behaviour of perturbations is determined by values of wavenumber of the carrier wave. We find, in particular, that for some range of wavenumbers of the carrier wave the waveguide is subjected to the local collapse which differs from collapse of waveguides in the fluid of infinite depth.

Discrete symmetry’s chains and links between integrable equations

We consider the discrete symmetry’s dressing chains of the nonlinear Schrödinger equation (NLS) and Davey–Stewartson equations (DS). The modified NLS (mNLS) equation and the modified DS (mDS) equations are obtained. The explicitly reversible Bäcklund auto-transformations for the mNLS and mDS equations are constructed. We demonstrate discrete symmetry’s conjugate chains of the KP and DS models. The two-dimensional generalization of the P4 equation is obtained.

Jacobi elliptic function solutions of nonlinear wave equations via the new sinh-Gordon equation expansion method

In this paper, based on the well-known sinh-Gordon equation, a new sinh-Gordon equation expansion method is developed. This method transforms the problem of solving nonlinear partial differential equations into the problem of solving the corresponding systems of algebraic equations. With the aid of symbolic computation, the procedure can be carried out by computer. Many nonlinear wave equations in mathematical physics are chosen to illustrate the method such as the KdV-mKdV equation, (2+1)-dimensional coupled Davey–Stewartson equation, the new integrable Davey–Stewartson-type equation, the modified Boussinesq equation, (2+1)-dimensional mKP equation and (2+1)-dimensional generalized KdV equation. As a consequence, many new doubly-periodic (Jacobian elliptic function) solutions are obtained. When the modulus m → 1 or 0, the corresponding solitary wave solutions and singly-periodic solutions are also found. This approach can also be applied to solve other nonlinear differential equations.

Ablowitz Ladik hierarchy and two-component Toda lattice hierarchy

It is shown in the language of free fermions and the τ functions that the Ablowitz–Ladik (AL) hierarchy and the relativistic Toda lattice hierarchy arise from the A(1)1 reduction of the two-component Toda lattice hierarchy. This result gives us a simple explanation of the reason why the AL hierarchy includes important integrable systems such as the discrete nonlinear Schrodinger equation, the relativistic Toda lattice equation, the Toda field equation, the Davey–Stewartson equation, etc. We also propose a new Backlund transformation (or a discretized time system) for the AL hierarchy.

Nonlinear wave focusing on water of finite depth

The problem of freak wave formation on water of finite depth is discussed. Dispersive focusing in a nonlinear medium is suggested as a possible mechanism of giant wave generation. This effect is considered within the framework of the nonlinear Schrödinger equation and the Davey–Stewartson system, describing 2+1-dimensional surface wave groups on water of finite depth. In the 2+1-dimensional case, the dispersive grouping is accompanied with a geometrical focusing. Necessary wave conditions for the occurrence of such a phenomenon are discussed. Influence of non-optimal phase modulation and presence of strong random wave component are found to be weak: they do not cancel the mechanism of wave amplification. The mechanism of dispersive focusing is compared with the wave enhancement due to the Benjamin–Feir instability, which is found to be extremely sensitive with respect to weak random perturbations.

QUANTUM INTEGRABLE SYSTEM WITH MULTI-COMPONENTS IN TWO DIMENSIONS

A quantum N-body problem with 2-component in (2 + 1) dimensions deduced from an integrable model in (2 + 1) dimensions is investigated. The Davey–Stewartson 1 (DS1) system9 is an integrable model in two dimensions. A quantum DS1 system with 2 color components in two dimensions has been formulated. This two-dimensional problem has been reduced to two one-dimensional many-body problems with 2 color components. The solutions of the two-dimensional problem under consideration has been constructed from the resulting problems in one dimension. For the latter with δ-function interactions and being solved by the Bethe–Yang ansatz, we introduce symmetrical and anti-symmetrical Young operators of the permutation group and obtain the exact solutions for the quantum DS1 system. The application of the solutions is discussed.