Davey Stewartson Research Articles

Interacting waves of Davey–Stewartson III system

Modulational unstable regions of the Davey–Stewartson (DS) III system have been determined from the generalized dispersion relation associating the frequency and wavenumber of the modulating perturbations. By means of the multilinear variable separation approach, the variable separation solution for the DS III equation is obtained with two arbitrary functions of (x,t) and two arbitrary functions of (y,t), which can be utilized to generate various (2+1)-dimensional localized excitations. Particular attention is paid on the interacting waves between periodic multivalued foldons and single-valued dromions, which can be viewed as periodic extensions of single foldon–dromion excitations.

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

AbstractIn this paper, the partially party‐time () symmetric nonlocal Davey–Stewartson (DS) equations with respect to x is called x‐nonlocal DS equations, while a fully symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x‐nonlocal DS equations, the usual (2 + 1)‐dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)‐dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.

Reverse Space-Time Nonlocal Sasa–Satsuma Equation and Its Solutions

The Sasa–Satsuma equation is an integrable high-order nonlinear Schrodinger (NLS) equation, and also is a complex modified KdV-type equation. It can describe the propagation of femtosecond pulses in optical fibers. Very recently, Ablowitz and Mussliman introduced a class of reverse space-time and reverse time nonlinear integrable equations, including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey–Stewartson equations, etc. So, what is nonlocal version of high-order NLS? In this paper, we introduce a reverse space-time nonlocal Sasa–Satsuma equation, i.e., a reverse space-time nonlocal high-order NLS equation, and derive its solutions with the binary Darboux transformation method. Periodic solutions, and some localized solutions, such as dark soliton, W-shaped soliton, M-shaped soliton and breather soliton of the reverse space-time nonlocal Sasa–Satsuma equation are constructed.

Energy scattering of a generalized Davey–Stewartson system in three dimension

In this paper, we study the global scattering result of the solution for the generalized Davey–Stewartson system $$\left\{ {\begin{array}{*{20}{c}} {i{\partial _t}u + \Delta u = {{\left| u \right|}^2}u + u{v_{{x_1}}},\left( {t,x} \right) \in \mathbb{R} \times {\mathbb{R}^3},} \\ { - \Delta u = {{\left( {{{\left| u \right|}^2}} \right)}_{{x_1}}}.} \end{array}} \right.$$ The main difficulties are the failure of the interaction Morawetz estimate and the asymmetrical structure of nonlinearity (in particular, the nonlinearity is non-local). To compensate, we utilize the strategy derived from concentration-compactness idea, which was first introduced by Kenig and Merle [Invent. Math., 166, 645–675 (2006)].

The residual symmetry and exact solutions of the Davey–Stewartson III equation

Through the truncated Painlevé expansion, the residual symmetry of the Davey–Stewartson III system is derived. This symmetry is localized to a properly prolonged system and the corresponding Bäcklund transformation is obtained. Based on the transformation, some exact solutions are generated. Furthermore, several types of interaction wave solutions are obtained by using the consistent tanh expansion method.

Applications of He's semi-inverse method, ITEM and GGM to the Davey-Stewartson equation

We investigate the Davey-Stewartson (DS) equation. Travelling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, He's semi-inverse variational principle method (SIVPM), the improved \(\tan(\phi/2)\)-expansion method (ITEM) and generalized \(G'/G\)-expansion method (GGM) for seeking more exact solutions via the DS equation. These methods are direct, concise and simple to implement compared to other existing methods. The exact solutions containing four types solutions have been achieved. The results demonstrate that the aforementioned methods are more efficient than the Ansatz method applied by Mirzazadeh (2015). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found by the improved \(\tan(\phi/2)\)-expansion and generalized \(G'/G\)-expansion methods. By He's semi-inverse variational principle we have obtained dark and bright soliton wave solutions. Also, the obtained semi-inverse variational principle has profound implications in physical understandings. These solutions might play important role in engineering and physics fields. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

New soliton solutions of Davey–Stewartson equation with power-law nonlinearity

This study is related to new soliton solutions of Davey–Stewartson equation (DSE) with power-law nonlinearity. The generalized Kudryashov method which is one of the analytical methods has been used for finding exact solutions of this equation. By using this method, dark soliton solutions of DSE have been found. Also, by using Mathematica Release 9, some graphical representations have been done to analyze the motion of these solutions.

Numerical study of blow-up to the purely elliptic generalized Davey–Stewartson system

Blow-up solutions for the purely elliptic generalized Davey–Stewartson system are studied by using a relaxation numerical method. The numerical method is based on an implicit finite-difference scheme with a second-order accuracy in both time and space. The stability of the numerical method is analyzed by investigating the linear stability of plane wave solutions. To evaluate the ability of the relaxation method to detect blow-up, numerical simulations are conducted for several test problems. A particular attention is paid to the gap interval neither a global existence nor a blow-up result is established. The monotonicity properties of blow-up time on the coupling parameter are also investigated numerically.

Modulation and nonlinear evolution of multi-dimensional Langmuir wave envelopes in a relativistic plasma

The modulational instability (MI) and the evolution of weakly nonlinear two-dimensional (2D) Langmuir wave (LW) packets are studied in an unmagnetized collisionless plasma with weakly relativistic electron flow. By using a 2D self-consistent relativistic fluid model and employing the standard multiple-scale technique, a coupled set of Davey-Stewartson (DS)-like equations is derived which governs the slow modulation and the evolution of LW packets in relativistic plasmas. It is found that the relativistic effects favor the instability of LW envelopes in the k{\theta} plane, where k is the wave number and {\theta} the angle of modulation. It is also found that as the electron thermal velocity or {\theta} increases, the growth rate of MI increases with cutoffs at higher wave numbers of modulation. Furthermore, in the nonlinear evolution of the DS-like equations, it is seen that with an effect of the relativistic flow, a Gaussian wave beam collapses in a finite time, and the collapse can be arrested when the effect of the thermal pressure or the relativistic flow is slightly relaxed. The present results may be useful to the MI and the formation of localized LW envelopes in cosmic plasmas with a relativistic flow of electrons.

The nonlinear dispersive Davey-Stewartson system for surface waves propagation in shallow water and its stability

The three-dimensional (3-D) nonlinear and dispersive PDEs system for surface waves propagating at undisturbed water surface under the gravity force and surface tension effects are studied. By applying the reductive perturbation method, we derive the (2 + 1) -dimensions form of the Davey-Stewartson (DS) system for the modulation of 2-D harmonic waves. By using the simplest equation method, we find exact traveling wave solutions and a general form of the multiple-soliton solution of the DS model. The dispersion analysis as well as the conservation law of the DS system are discussed. It is revealed that the consistency of the results with the conservation of the potential energy increases with increasing Ursell parameter. Also, the stability of the ODEs form of the DS system is presented by using the phase portrait method.

Three-dimensional lattice of Bäcklund transformations of integrable cases of the Davey–Stewartson system

We construct a three-dimensional octahedral lattice of Backlund transformations of integrable cases of the Davey–Stewartson system. At the lattice sites, we arrange functions, which, on one hand, are used to define the dynamical variables of the Davey–Stewartson system and, on the other hand, are connected by bilinear relations of the Hirota type. One of the lattice equations is a purely discrete six-point equation that coincides with the famous Hirota equation.

Boundary Conditions for Infinite Conservation Laws

Regular soliton equations (KdV, sine-Gordon, NLS) are known to possess infinite sets of local conservation laws. Some other classes of nonlinear PDE possess infinite-dimensional symmetries parametrized by arbitrary functions of independent or dependent variables; among them are Zabolotskaya–Khokhlov, Kadomtsev–Petviashvili, Davey–Stewartson equations and Born-Infeld equation. Boundary conditions were shown to play an important role for the existence of local conservation laws associated with infinite-dimensional symmetries. In this paper, we analyze boundary conditions for the infinite conserved densities of regular soliton equations: KdV, potential KdV, Sine-Gordon equation, and nonlinear Schrodinger equation, and compare them with boundary conditions for the conserved densities obtained from infinite-dimensional symmetries with arbitrary functions of independent and dependent variables.

Integrable Nonlocal Nonlinear Equations

A nonlocal nonlinear Schrödinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced “potential” is symmetric thus the nonlocal NLS equation is also symmetric. In this paper, new reverse space‐time and reverse time nonlocal nonlinear integrable equations are introduced. They arise from remarkably simple symmetry reductions of general AKNS scattering problems where the nonlocality appears in both space and time or time alone. They are integrable infinite dimensional Hamiltonian dynamical systems. These include the reverse space‐time, and in some cases reverse time, nonlocal NLS, modified Korteweg‐deVries (mKdV), sine‐Gordon, (1 + 1) and (2 + 1) dimensional three‐wave interaction, derivative NLS, “loop soliton,” Davey–Stewartson (DS), partially symmetric DS and partially reverse space‐time DS equations. Linear Lax pairs, an infinite number of conservation laws, inverse scattering transforms are discussed and one soliton solutions are found. Integrable reverse space‐time and reverse time nonlocal discrete nonlinear Schrödinger type equations are also introduced along with few conserved quantities. Finally, nonlocal Painlevé type equations are derived from the reverse space‐time and reverse time nonlocal NLS equations.

Global well-posedness and long-time asymptotics for the defocussing Davey–Stewartson II equation in $H^{1,1}(\mathbb C)$

We use the \bar {\partial} -inverse scattering method to obtain global well-posedness and large-time asymptotics for the defocussing Davey–Stewartson II equation. We show that these global solutions are dispersive by computing their leading asymptotic behavior as t \to \infty in terms of an associated linear problem. These results appear to be sharp.

Optical soliton wave solutions to the resonant Davey–Stewartson system

We investigate the resonant Davey–Stewartson (DS) system. The resonant DS system is a natural \((2+1)\)-dimensional version of the resonant nonlinear Schrodinger equation. Traveling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, improved \(\tan (\phi /2)\)-expansion method (ITEM) and generalized (G′/G)-expansion method for seeking more exact solutions via the resonant Davey–Stewartson system. These methods are direct, concise and simple to implement compared to other existing methods. The exact particular solutions containing four types of solutions, i.e., hyperbolic function, trigonometric function, exponential and solutions. We obtained further solutions comparing these methods with other methods. The results demonstrate that the aforementioned methods are more efficient than the multi-linear variable separation method applied by Tang et al. (Chaos Solitons Fractals 42:2707–2712, 2009). Recently the ITEM was developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important role in engineering and physics fields. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving the nonlinear problems.

New acoustic wave behaviors to the Davey–Stewartson equation with power-law nonlinearity arising in fluid dynamics

This manuscript investigates the new acoustic wave behaviors of the Davey–Stewartson equation with power-law nonlinearity with the help of sine-Gordon expansion method. This technique yields many new acoustic wave behaviors such as hyperbolic, exponential and complex function structures to the problem considered. Wolfram Mathematica 9 has been used throughout the paper for mathematical calculations along with obtaining two- and three-dimensional surfaces of results.

A New Reduction of the Self-Dual Yang–Mills Equations and its Applications

Abstract Through imposing on space–time symmetries, a new reduction of the self-dual Yang–Mills equations is obtained for which a Lax pair is established. By a proper exponent transformation, we transform the Lax pair to get a new Lax pair whose compatibility condition gives rise to a set of partial differential equations (PDEs). We solve such PDEs by taking different Lax matrices; we develop a new modified Burgers equation, a generalised type of Kadomtsev–Petviasgvili equation, and the Davey–Stewartson equation, which also generalise some results given by Ablowitz, Chakravarty, Kent, and Newman.

Action of a scattering map on weighted Sobolev spaces in the plane

We consider a scattering map that arises in the ∂¯ approach to the scattering theory for the Davey–Stewartson II equation and show that the map is an invertible map between certain weighted L2 Sobolev spaces.

Periodic solitons for the elliptic–elliptic focussing Davey–Stewartson equations

We consider the elliptic–elliptic, focussing Davey–Stewartson equations, which have an explicit bright line soliton solution. The existence of a family of periodic solitons, which have the profile of the line soliton in the longitudinal spatial direction and are periodic in the transverse spatial direction, is established using dynamical systems arguments. We also show that the line soliton is linearly unstable with respect to perturbations in the transverse direction.

Bilinearization and new multisoliton solutions for the (4+1)-dimensional Fokas equation

The (4 + 1)-dimensional Fokas equation is derived in the process of extending the integrable Kadomtsev–Petviashvili and Davey–Stewartson equations to higher-dimensional nonlinear wave equations. This equation is under investigation in this paper. Hirota’s bilinear method is, for the first time, used to solve such a higher-dimensional equation. In order to bilinearize the Fokas equation, some appropriate transformations are adopted. As a result, single-soliton solution, double-soliton solution and three-soliton solution are obtained. A new uniform formula of n-soliton solution is derived from this. It is shown that the transformations adopted in this work play a key role in converting the Fokas equation into Hirota’s bilinear form.