Davey-Stewartson Equation Research Articles

New soliton wave structures of nonlinear (4 + 1)-dimensional Fokas dynamical model by using different methods

The physics of the (4 + 1)-dimensional Fokas equation follows necessarily from the physical nature of the Kadomtsev–Petviashvili and Davey-Stewartson equations in wave theory. In this article, we consider the nonlinear (4+1)-dimensional Fokas partial differential equation. Two recognizable methods namely improved F-expansion and generalized exp(-ϕ(ξ))-expansion methods are proposed to investigate the new wave structures of (4 + 1)-dimensional Fokas dynamical model which have never been constructed before. As a result, new solutions are in different solitons such as dark soliton, bright soliton, combined dark-bright soliton solutions, periodic, and solitary wave solutions. The new generalized solitary wave solutions can be constructed by assigning the specific values to those parameters involved in these methods. The achieved results are compared with existing results in the literature and we found that some of results are not available in literature. The derived results are explained graphically, to understand the phenomena of the proposed model. The constructed results rendering that the consider methods in this article are simple, effective, and easy to find the solution of many other nonlinear higher-dimensional models which arise in several areas of science and engineering.

High precision numerical approach for Davey–Stewartson II type equations for Schwartz class initial data

We present an efficient high-precision numerical approach for Davey–Stewartson (DS) II type equa- tions, treating initial data from the Schwartz class of smooth, rapidly decreasing functions. As with previous approaches, the presented code uses discrete Fourier transforms for the spatial dependence and Driscoll’s composite Runge–Kutta method for the time dependence. Since DS equations are non-local, nonlinear Schrödinger equations with a singular symbol for the non-locality, standard Fourier methods in practice only reach accuracy of the order of 10−6or less for typical examples. This was previously demonstrated for the defocusing integrable case by comparison with a numerical approach for DS II via inverse scattering. By applying a regularization to the singular symbol, originally developed for D-bar problems, the presented code is shown to reach machine precision. The code can treat integrable and non-integrable DS II equations. Moreover, it has the same numerical complexity as existing codes for DS II. Several examples for the integrable defocusing DS II equation are discussed as test cases. In an appendix by C. Kalla, a doubly periodic solution to the defocusing DS II equation is presented, providing a test for direct DS codes based on Fourier methods.

On the instability of ground states for a generalized Davey-Stewartson system

In this paper, we give a simpler proof for Ohta’s theorems [1995, Ann. Inst. Henri Poincare, 63, 111; 1995, Diff. Integral Eq., 8, 1775] on the strong instability of the ground states for a generalized Davey-Stewartson system. In addition, a sufficient condition is given to ensure the nonexistence of a minimizer for a variational problem, which is related to the stability of the standing waves of the Davey-Stewartson system. This result shows that the stability result of Ohta [Diff. Integral Eq., 8, 1775] is sharp.

Solitary and traveling wave solutions for the Davey–Stewartson equation using the Jacobi elliptic function expansion method

In our paper we modify the Jacobi elliptic function expansion method to obtain solutions to the Davey–Stewartson system of equations. Two categories of nonsingular solutions are obtained for both traveling and solitary waves and both with and without chirp. In both cases there is an arbitrary term in the mean flow field, meaning one can obtain solutions for arbitrary forms of the mean flow field.

On the exact solutions to some system of complex nonlinear models

Abstract In this manuscript, the application of the extended sinh-Gordon equation expansion method to the Davey-Stewartson equation and the (2+1)-dimensional nonlinear complex coupled Maccari system is presented. The Davey-Stewartson equation arises as a result of multiple-scale analysis of modulated nonlinear surface gravity waves propagating over a horizontal seabed and the (2+1)-dimensional nonlinear complex coupled Maccari equation describes the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics. We successfully construct some soliton, singular soliton and singular periodic wave solutions to these two nonlinear complex models. The 2D, 3D and contour graphs to some of the obtained solutions are presented.

Periodic-wave and semirational solutions for the (2 $$+$$ 1)-dimensional Davey–Stewartson equations on the surface water waves of finite depth

The (2 $$+$$ 1)-dimensional Davey–Stewartson equations concerning the evolution of surface water waves with finite depth are studied. We derive the periodic-wave solutions through the Kadomtsev–Petviashvili hierarchy reduction. We obtain the growing-decaying periodic wave and three kinds of breathers via those solutions. We obtain the periodic wave takes on the growing and decaying property. Taking the long-wave limit on the periodic-wave solutions, we derive the semirational solutions describing the interaction of the rogue wave, lump, breather and periodic wave. We illustrate the lump and rogue wave and find that the rogue wave (lump) is the long-wave limit of the periodic wave (breather).

Reduction in the $$\mathbf {(4+1)}$$-dimensional Fokas equation and their solutions

An integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations is investigated in this paper. We will refer to this integrable extension as the $$(4+1)$$-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the $$(4+1)$$-dimensional Fokas equation are constructed based on the Hirota’s bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the $$(1+1)$$-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of $$(3+1)$$-dimensional KP equation by reducing the solutions of the $$(4+1)$$-dimensional Fokas equation. All these results show that the $$(4+1)$$-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, nonlinear optics, and photonics, ion-acoustic waves in plasmas, matter waves in Bose–Einstein condensates, and sound waves in ferromagnetic media.

Davey–Stewartson equations in (3 + 1) dimensions with an infinite-dimensional symmetry algebra

This article is devoted to discovering Lie symmetry algebra of a (3+1)-dimensional Davey-Stewartson system which appears in the field of plasma physics. It is found that the algebra is an infinite dimensional one and of Kac-Moody type. Making use of these symmetries, some reduced equations to lower dimensions are also presented.

Novel solitons and periodic wave solutions for Davey–Stewartson system with variable coefficients

In this paper, the variable coefficients Davey–Stewartson system represents many physical phenomena in shallow water waves, quantum and optics, etc, is transformed directly into nonlinear ordinary differential system by using the new modification to the direct similarity reduction method. After solving the reduced system, new Jacobi, hyperbolic and periodic wave solutions are achieved for complex variable coefficients Davey–Stewartson system. The application of the new modification of the direct similarity reduction method reflects how this method is powerful, easy and simple, if it is compared with other symmetry techniques.

New Weierstrass elliptic wave solutions of the Davey–Stewartson equation with power law nonlinearity

New Weierstrass elliptic wave solutions of the Davey–Stewartson equation with power law nonlinearity

CAUCHY PROBLEM FOR THE GENERALIZED DAVEY-STEWARTSON SYSTEMS IN BESOV SPACES AND SOME COUNTEREXAMPLES

In this paper, the Cauchy problem of the generalized ellipse-ellipse type Davey-Stewartson systems is discussed. When the dimension of space is greater than or equal to two, we get a unique global solution in Besov spaces by contraction mapping argument. Moreover, by using the F-expansion method, the exact periodic wave solutions for the generalized ellipse-ellipse type DaveyStewartson systems are discussed, some counter examples are given.

Constructing new wave solutions to the $$(2 + 1)$$-dimensional Davey–Stewartson equation (DSE) which arises in fluid dynamics

In this paper, we utilize the uniform algebraic method and construct some new wave solutions to the $$(2 + 1)$$-dimensional Davey–Stewartson equation which arises in fluid dynamics. We successfully obtain some hyperbolic and trigonometric function solutions to this equation. Subsequently, We have also built the Lagrangian and the Hamiltonian for the second-order nonlinear ODE corresponding to the traveling wave reduction of the (DSE)-equation. Moreover, we plot 2D and 3D surfaces representing the obtained solutions by considering some suitable values of the parameters. The unique (2 + 1)-dimensional dynamics of the obtained exact solutions of the (DSE) equation may help to understand the formation and key properties of traveling waves in many other physical settings. Before the end of this paper, we compare our results with the existing results in the literature by presenting comprehensive conclusions.

Solitary wave solutions for the variable-coefficient coupled nonlinear Schrödinger equations and Davey–Stewartson system using modified sine-Gordon equation method

In this study, the sine-Gordon equation method is modified to deal with variable-coefficient systems containing imaginary parts, such as nonlinear Schrödinger systems. These are of considerable importance in many fields of research, including ocean engineering and optics. As an example, we apply the modified method to variable-coefficient coupled nonlinear Schrö dinger equations and Davey–Stewartson system with variable coefficients, treating them as one-dimensional and two-dimensional systems, respectively. As a result of this application, novel solitary wave solutions are obtained for both cases. Moreover, some figures are provided to illustrate how the solitary wave propagation is determined by the different values of the variable group velocity dispersion terms, which can be used to model various phenomena.

A nonlinear Plancherel theorem with applications to global well-posedness for the defocusing Davey–Stewartson equation and to the inverse boundary value problem of Calderón

We prove a Plancherel theorem for a nonlinear Fourier transform in two dimensions arising in the Inverse Scattering method for the defocusing Davey–Stewartson II equation. We then use it to prove global well-posedness and scattering in $$L^2$$ for defocusing DSII. This Plancherel theorem also implies global uniqueness in the inverse boundary value problem of Calderon in dimension 2, for conductivities $$\sigma >0$$ with $$\log \sigma \in \dot{H}^1$$. The proof of the nonlinear Plancherel theorem includes new estimates on classical fractional integrals, as well as a new result on $$L^2$$-boundedness of pseudo-differential operators with non-smooth symbols, valid in all dimensions.

Traveling wave solutions for the (3+1)-dimensional Davey-Stewartson equations

In this work, the extended tanh method is used to construct wave solutions for the Davey–Stewartson equations. The extended tanh method is a powerful solution method for obtaining different kind solutions of nonlinear evolution equations. This method can be applied to nonintegrable equations as well as to integrable ones.

On the two-component generalization of the (2+1)-dimensional Davey-Stewartson I equation

The geometric-gauge equivalent of the famous Ishimori spin equation is the (2+1)-dimensional Davey-Stewartson equation, which in turn is one of the (2+1)-dimensional generalizations of the nonlinear Schrodinger equation. Multicomponent generalization of nonlinear integrable equations attract considerable interest from both physical and mathematical points of view. In this paper, the two-component integrable generalization of the (2+1)-dimensional Davey-Stewartson I equation is obtained based on its one-component representation, and the corresponding Lax representation is also obtained.

Nonintegrable spatial discrete nonlocal nonlinear schrödinger equation.

Integrable and nonintegrable discrete nonlinear Schrödinger equations (NLS) are significant models to describe many phenomena in physics. Recently, Ablowitz and Musslimani introduced a class of reverse space, reverse time, and reverse space-time nonlocal integrable equations, including the nonlocal NLS equation, nonlocal sine-Gordon equation, nonlocal Davey-Stewartson equation, etc. Moreover, the integrable nonlocal discrete NLS has been exactly solved by inverse scattering transform. In this paper, we study a nonintegrable discrete nonlocal NLS, which is a direct discrete version of the reverse space nonlocal NLS. By applying discrete Fourier transform and modified Neumann iteration, we present its stationary solutions numerically. The linear stability of the stationary solutions is examined. Finally, we study the Cauchy problem for the nonlocal NLS equation numerically and find some different and new properties on the numerical solutions comparing with the numerical solutions of the Cauchy problem for the NLS equation.

On the dispersionless Davey–Stewartson system: Hamiltonian vector field Lax pair and relevant nonlinear Riemann–Hilbert problem for dDS-II system

In this paper, the semiclassical limit of Davey-Stewartson systems are studied. It shows that these dispersionless limited integrable systems of hydrodynamic type, which are defined as dDS (dispersionless Davey-Stewartson) systems, are arisen from the commutation condition of Lax pairs of one-parameter vector fields. The relevant nonlinear Riemann-Hilbert problems with some symmetries for the dDS-II system are also constructed. This kind of Riemann-Hilbert problems are meaningful for applying the formal inverse scattering transform method, recently developed by Manakov and Santini, to study the dDS-II system .

New Exact Solutions and Modulation Instability for the Nonlinear (2+1)-Dimensional Davey-Stewartson System of Equation

The Davey-Stewartson Equation (DSE) is an equation system that reflects the evolution in finite depth of soft nonlinear packets of water waves that move in one direction but in which the waves’ amplitude is modulated in spatial directions. This paper uses the Generalized Elliptic Equation Rational Expansion (GEERE) technique to extract fresh exact solutions for the DSE. As a consequence, solutions with parameters of trigonometric, hyperbolic, and rational function are achieved. To display the physical characteristics of this model, the solutions obtained are graphically displayed. Modulation instability assessment of the outcomes acquired is also discussed and it demonstrates that all the solutions built are accurate and stable.

Many New Exact Solutions for Generalized Davey-Stewartson Equation with Arbitrary Power Nonlinearities Using Novel (G'/G)-Expansion Method

Many New Exact Solutions for Generalized Davey-Stewartson Equation with Arbitrary Power Nonlinearities Using Novel (G'/G)-Expansion Method