Davey Stewartson Research Articles

On a (2 + 1)-dimensional generalization of the Ablowitz–Ladik lattice and a discrete Davey–Stewartson system

We propose a natural (2 + 1)-dimensional generalization of the Ablowitz–Ladik lattice that is an integrable space discretization of the cubic nonlinear Schrödinger system in 1 + 1 dimensions. By further requiring rotational symmetry of order 2 in the two-dimensional lattice, we identify an appropriate change of dependent variables, which translates the (2 + 1)-dimensional Ablowitz–Ladik lattice into a suitable space discretization of the Davey–Stewartson system. The space-discrete Davey–Stewartson system has a Lax pair and allows the complex conjugation reduction between two dependent variables as in the continuous case. Moreover, it is ideally symmetric with respect to space reflections. Using the Hirota bilinear method, we construct some exact solutions such as multidromion solutions.

On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

In this paper, we derive a class of doubly periodic standing wave solutions for a coupled Higgs field equation by employing the Hirota bilinear method and theta function identities. Such solutions are expressed in terms of theta functions with variable separation form. Moreover, it is shown that these solutions can be converted into Jacobi elliptic function representations, and their long‐wave limit yields collision of dark solitons. In comparing with known solutions of the canonical evolution equation, three new aspects will be developed in this paper. First, the periods in the spatial and temporal directions, measured in terms of the theta function parameters τ and τ1, are independent of each other, quite unlike most similar solutions found earlier in the literature. Second, the doubly periodic wave solutions possess two families of the local maxima, whose locations and types are then examined in detail. Third, we obtain new doubly periodic standing wave solutions for the Davey–Stewartson equation through its similarity transformation to the coupled Higgs field equation.

Full Symmetry Groups and Similar Reductions of a (2+1)-Dimensional Resonant Davey—Stewartson System

Applying the classical Lie symmetry method to the (2+1)-dimensional resonant Davey—Stewartson system introduced by Tang [X.Y. Tang et al., Chaos, Solitons and Fractals 42 (2007) 2707], a more general infinite dimensional Lie symmetry with Kac-Moody-Virasoro type Lie algebra is obtained, which involves four arbitrary functions of t. Alternatively, by a simple direct method, the full symmetry groups including Lie symmetry group and non-Lie symmetry group are gained straightly. In this way, the related Lie algebra can be easily found by a more simple limiting procedure. Lastly, via solving the characteristic equations, three types of the general similar reductions are derived.

Quasi-line soliton interactions of the Davey–Stewartson I equation: on the existence of long-range interaction between two quasi-line solitons through a periodic soliton

A periodic soliton is turned into a line soliton accordingly as a parameter point approaches to the boundary of the existing domain in the parameter space for a nonsingular periodic-soliton solution. We will call the periodic soliton with parameters of the neighborhood of the boundary a quasi-line soliton in this paper, which seems to be the line soliton. The interaction between two quasi-line solitons is the same as the interaction between two line solitons, except for very small parameter-sensitive regions. However, in such parameter regions, there are new long-range interactions between two quasi-line solitons through the periodic soliton as the messenger under some conditions, which cannot be described by the two-line-soliton solution.

Numerical simulation of blow-up solutions for the generalized Davey–Stewartson system

Blow-up solutions for the generalized Davey–Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed.

On a 2+1-Dimensional Whitham-Broer-Kaup System: A Resonant NLS Connection

It is established that the Whitham–Broer–Kaup shallow water system and the “resonant” nonlinear Schrodinger equation are equivalent. A symmetric integrable 2+1-dimensional version of the Whitham–Broer–Kaup system is constructed which, in turn, is equivalent to a recently introduced resonant Davey–Stewartson I system incorporating a Madelung–Bohm type quantum potential. A bilinear representation is adopted and resonant solitonic interaction in this new 2+1-dimensional Kaup–Broer system is exhibited.

Fourth Order Time-Stepping for Kadomtsev–Petviashvili and Davey–Stewartson Equations

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 combination with spectral methods is beneficial in resolving the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kadomtsev–Petviashvili and the Davey–Stewartson equations, two integrable equations in $2+1$ dimensions: these methods are exponential time-differencing, integrating factors, time-splitting, implicit Runge–Kutta, and Driscoll's composite Runge–Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.

The Kadomtsev–Petviashvili II equation on the half-plane

The KPII equation is an integrable nonlinear PDE in 2+1 dimensions (two spatial and one temporal), which arises in several physical circumstances, including fluid mechanics, where it describes waves in shallow water. It provides a multidimensional generalisation of the renowned KdV equation. In this work, we employ a novel approach recently introduced by one of the authors in connection with the Davey–Stewartson equation (Fokas (2009) [13]), in order to analyse the initial-boundary value problem for the KPII equation formulated on the half-plane. The analysis makes crucial use of the so-called d -bar formalism, as well as of the so-called global relation. A novel feature of boundary as opposed to initial value problems in 2+1 is that the d -bar formalism now involves a function in the complex plane which is discontinuous across the real axis.

The sharp threshold and limiting profile of blow-up solutions for a Davey–Stewartson system

The blow-up solutions of the Cauchy problem for the Davey–Stewartson system, which is a model equation in the theory of shallow water waves, are investigated. Firstly, the existence of the ground state for the system derives the best constant of a Gagliardo–Nirenberg type inequality and the variational character of the ground state. Secondly, the blow-up threshold of the Davey–Stewartson system is developed in R 3 . Thirdly, the mass concentration is established for all the blow-up solutions of the system in R 2 . Finally, the existence of the minimal blow-up solutions in R 2 is constructed by using the pseudo-conformal invariance. The profile of the minimal blow-up solutions as t → T (blow-up time) is in detail investigated in terms of the ground state.

Solution of the Davey–Stewardson equation using homotopuy analysis method

In this paper, the homotopy analysis method (HAM) proposed by Liao is adopted for solving Davey–Stewartson (DS) equations which arise as higher dimensional generalizations of the nonlinear Schrödinger (NLS) equation. The results obtained by HAM have been compared with the exact solutions and homotopy perturbation method (HPM) to show the accuracy of the method. Comparisons indicate that there is a very good agreement between the HAM solutions and the exact solutions in terms of accuracy.

The [formula omitted] method and 1-soliton solution of the Davey–Stewartson equation

This paper studies the Davey–Stewartson equation. The G′G method is applied to carry out the integration of this equation. Subsequently, using the ansatz method this equation is integrated in (1+2) dimensions with power law nonlinearity.

Geometric optics and instability for NLS and Davey–Stewartson models

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schr¨odinger equations with gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey–Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev spaces, where we prove norm inflation with infinite loss of regularity by a constructive approach.

Heteroclinic Breather-Wave Solutions for Davey–Stewartson Equation

Exact heteroclinic breather-wave solutions for Davey–Stewartson (DSI, DSII) system with periodic boundary condition are constructed using Hirota's bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.

A Hamiltonian approach to nonlinear modulation of surface water waves

Many wave phenomena in physics are described by weakly nonlinear nearly monochromatic solutions in the form of modulated wave packets. The examples include ocean waves as well as waves in optics and in plasmas. There are a number of approaches to deriving the envelope equations for these theories of amplitude modulation. In this paper, we give a unified approach, based on the principles of a Hamiltonian formulation of the equations of motion. Our principal example is the system of equations of free surface water waves, for which we give a new derivation of the classical nonlinear Schrödinger and Davey–Stewartson equations, as well as the higher-order Dysthe system. One consequence of our analysis from this point of view is that the Dysthe equation can be posed as a Hamiltonian partial differential equation.

Long-Range Interaction between Two Periodic Solitons through Growing-and-Decaying Mode in the Davey–Stewartson I Equation

where r is constant. One important feature of the solutions to the DS I equation is the reversion of the unstable wave train to its initial state. The analytical solution for the nonlinear evolution of a modulational unstable mode is given by the growing-and-decaying (GD) mode solution. This shows that an unstable mode grows exponentially in its early stage. After reaching a maximummodulation, the unstable mode vanishes with time to reproduce the initial unmodulated state. The DS equation has a periodic soliton solution, which represents the spatial structure of the inclined sequence of algebraic solitons in addition to the dark line soliton and algebraic soliton solutions. In previous studies, we investigated the interactions between two periodic solitons, and between the periodic soliton and other types of solitons. Note that there are two types of singular interaction: the resonant interaction, in which two solitons interact so as to generate a single soliton, and the extremely long-range interaction, in which two solitons interchange each other infinitely apart through the messenger soliton, which is similar to a periodic soliton. Recently, the interaction between a soliton and a GD mode has been investigated, and the existence of resonant interactions has been demonstrated. We reported that the existence of the periodic soliton changed the evolution of the modulational instability markedly, as if the soliton dominated the evolution of instability. These results suggest the possible existence of an interaction between two solitons through the GD mode. The purpose of the present note is to show that a long-range interaction exists between two parallel periodic solitons through the GD mode. Using the N-soliton solution of Satsuma and Ablowitz, the solution that describes the interaction between two periodic solitons, which have the same real part ð ; Þ and different imaginary parts ð 1; 1Þ and ð 2; 2Þ of complex wave numbers, is written as

CLASSIFICATION OF ALL ENVELOPE TRAVELING WAVE SOLUTIONS TO (2+1)-DIMENSIONAL DAVEY–STEWARTSON EQUATION

Using the complete discrimination system for polynomial method, we study the (2+1)-dimensional Davey–Stewartson equation and obtain the classifications of all its envelope traveling wave solutions. This is very convenient in practice to give the corresponding solutions for the concrete parameters.

The Cauchy problem for the elliptic–hyperbolic Davey–Stewartson system in Sobolev space

We study the initial value problem for the elliptic–hyperbolic Davey–Stewartson systems (0.1) { i ∂ t u + Δ u = c 1 | u | 2 u + c 2 u ∂ x 1 φ , ( t , x ) ∈ R 3 , ( ∂ x 1 2 − ∂ x 2 2 ) φ = ∂ x 1 | u | 2 , u ( 0 , x ) = ϕ ( x ) , where Δ = ∂ x 1 2 + ∂ x 2 2 , c 1 , c 2 ∈ R , u is a complex valued function and φ is a real valued function. Our purpose is to prove the local existence and uniqueness of the solution for (0.1) in the Sobolev space H 3 / 2 + ( R 2 ) with small mass. Our methods rely heavily on Hayashi and Hirata (1996) [11], but we improve partial results of it, which got global existence of small solutions to (0.1) in weighted Sobolev space H 3 , 0 ∩ H 0 , 3 . Our main new tools are Kenig–Ponce–Vega type commutator estimate in Kenig, Ponce and Vega (1993) [16] and its variant form.

On conserved densities and boundary conditions for the Davey–Stewartson equations

In the paper we discuss local conservation laws with non-vanishing conserved densities and corresponding boundary conditions for the Davey–Stewartson equations. Proceeding with an infinite symmetry algebra of the Davey–Stewartson system we generate a finite number of conserved quantities through appropriate asymptotic conditions.

Applications of the New Compound Riccati Equations Rational Expansion Method and Fan's Subequation Method for the Davey-Stewartson Equations

We used what we called extended Fan's sub-equation method and a new compound Riccati equations rational expansion method to construct the exact travelling wave solutions of the Davey-Stewartson (DS) equations. The basic idea of the proposed extended Fan's subequation method is to take fulls advantage of the general elliptic equations, involving five parameters, which have many new solutions and whose degeneracies lead to special subequations involving three parameters like Riccati equation, first-kind elliptic equation, auxiliary ordinary equation and generalized Riccati equation. Many new exact solutions of the Davey-Stewartson (DS) equations including more general soliton solutions, triangular solutions, and double-periodic solutions are constructed by symbolic computation.

Painlevé analysis and exact solutions of the resonant Davey–Stewartson system

It is shown that the resonant Davey–Stewartson (RDS) system can pass the Painlev test. By truncating the Laurent series to a constant level term, a dependent variable transformation is naturally derived, which leads to the bilinear forms of the RDS system. From the bilinear equations, through making suitable assumptions, some new soliton solutions are obtained. Some representative profiles of the solitary waves are graphically displayed including the two-line soliton solution, “Y” soliton solution, “V” soliton solution, solitoff, etc. The solutions might be useful to describe the nonlinear phenomena in Madelung fluids, capillarity fluids, and so on.