Dimensional Davey-Stewartson Equation Research Articles

Dynamics of novel soliton and periodic solutions to the coupled fractional nonlinear model

This study secures the soliton solutions of the (2+1)-dimensional Davey–Stewartson equation (DSE) incorporating the properties of the truncated M-fractional derivative. The DSE and its coupling with other systems have extensive applications in many fields, including physics, applied mathematics, engineering, hydrodynamics, plasma physics, and nonlinear optics. Various solutions, such as dark, singular, bright-dark, bright, complex, and combined solitons, are derived. In addition, exponential, periodic, and hyperbolic solutions are also generated. The newly designed integration method, known as the modified Sardar subequation method (MSSEM), has been applied in this study for extracting the solutions. The approach is efficient in explaining fractional nonlinear partial differential equations (FNLPDEs) by confirming pre-existing solutions and producing new ones. Furthermore, we plot the density, 2D, and 3D graphs with the associated parameter values to visualize the solutions. The outcomes of this work indicate the effectiveness of the method utilized to improve nonlinear dynamical behavior. We anticipate that our work will be helpful for a large number of engineering models and other related problems.

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).

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.

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.

Solitons and Conservation Laws for the (2+1)-Dimensional Davey-Stewartson Equations with Conformable Derivative

Solitons and Conservation Laws for the (2+1)-Dimensional Davey-Stewartson Equations with Conformable Derivative

On the exact solutions of the fractional (2+1)- dimensional Davey - Stewartson equation system

In this work, we construct the exact traveling wave solutions of the fractional (2+1)-dimensional Davey-Stewartson equation system (D-S) that is complex equation system using the Modified Trial Equation Method (MTEM). We obtained trigonometric function solutions by this method that are new in literature.

Exact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized $(\frac{G'}{G})$-Expansion Method

In this article, we construct the exact traveling wave solutions of the nonlinear (2+1)-dimensional Davey-Stewartson equation (D-S) using the generalized $(\frac{G'}{G})$-expansion method which play an important role in mathematical physics. As a result, hyperbolic, trigonometric and rational function solutions with parameters are obtained. When these parameters are taken special values, the solitary and periodic solutions are derived from the hyperbolic and trigonometric function solutions respectively. New complex type traveling wave solutions to the nonlinear (2+1)-dimensional Davey-Stewartson equation were obtained with Liu's theorem.

New Solitons and Periodic Solutions for (2+1)-dimensional Davey-Stewartson Equations

In this paper, we find exact solutions for the (2+1)-dimensional Davey-Stewartson (DS) equations. The sine-cosine, tanh-coth, and exp-function methods are used to construct periodic and soliton solutions of these equations. Many new families of exact solutions of the DS equations are successfully obtained. These solutions may be of significant importance for the explanation of some practical physical problems.

Two Integral Methods Applied to the (2+1) Dimensional Davey-Stewartson Equation

In this paper, we use two integral methods, the first integral method and the direct integral method to study (2+1)- dimensional Davey-Stewartson equation . The first integral method was used to construct travelling wave solutions, those solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. By using the direct integration method shock wave solution and Jacobi elliptic function solutions are obtained. By comparison be- tween the two methods, the direct integration is more impressive than the first integral method. The results obtained con- firm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear systems of par- tial differential equations.

Solitary Wave Solution of the Two-Dimensional Regularized Long-Wave and Davey-Stewartson Equations in Fluids and Plasmas

This paper investigates the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2+1) dimensional Davey-Stewartson (DS) equation which is governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice. By using extended mapping method technique, we have shown that the 2DRLG-2DDS equations can be reduced to the elliptic-like equation. Then, the extended mapping method is used to obtain a series of solutions including the single and the combined non degenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear partial differential equations (NLPDEs).

Exact propagating dromion-like localized wave solutions of generalized ( [formula omitted])-dimensional Davey–Stewartson equations

The capacity of the double-Exponential (Exp) function method as an alternative approach to obtain the analytic solutions of higher dimensional coupled nonlinear partial differential equations in Engineering mathematics has been revealed. We employ the double-Exp method aided with symbolic computation to construct dromion-like localized solutions for a ( 2 + 1 ) -dimensional generalized Davey–Stewartson (2DGDS) system of equations governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multi-dimensional lattice. The multidromion-like solutions obtained by the present approach are more general than those obtained earlier.

Exact solutions and excitations for the Davey-Stewartson equations with nonlinear and gain terms

We study the general (2+1)-dimensional Davey-Stewartson (DS) equations with nonlinear and gain terms and acquire explicit solutions through variable separation approach. In particular, we deduce some main novel excitations for the DS equations, and further demonstrate different features of these excitations. More importantly, the similar solutions and excitations can be predicted to exist in other related revolution equations such as nonlinear Schr\"odinger equation to explain the Bose-Einstein condensation.

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.

New Types of Travelling Wave Solutions From (2+1)-DimensionalDavey–Stewartson Equation

In this paper, based on new auxiliary nonlinear ordinarydifferential equation with a sixth-degree nonlinear term, westudy the (2+1)-dimensional Davey–Stewartson equation and newtypes of travelling wave solutions are obtained, which includenew bell and kink profile solitary wave solutions, triangularperiodic wave solutions, and singular solutions. The method usedhere can be also extended to many other nonlinear partial differentialequations.

Construction of a series of travelling wave solutions to nonlinear equations

In this paper, based on new auxiliary ordinary differential equation with a sixth-degree nonlinear term, we study the (1 + 1)-dimensional combined KdV–MKdV equation, Hirota equation and (2 + 1)-dimensional Davey–Stewartson equation. Then, a series of new types of travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.

Complex hyperbolic-function method and its applications to nonlinear equations

Based on computerized symbolic computation, a complex hyperbolic-function method is proposed for the general nonlinear equations of mathematical physics in a unified way. In this method, we assume that exact solutions for a given general nonlinear equations be the superposition of different powers of the sech-function, tanh-function and/or their combinations. After finishing some direct calculations, we can finally obtain the exact solutions expressed by the complex hyperbolic function. The characteristic feature of this method is that we can derive exact solutions to the general nonlinear equations directly without transformation. Some illustrative equations, such as the ( 1 + 1 )-dimensional coupled Schrödinger–KdV equation, ( 2 + 1 )-dimensional Davey–Stewartson equation and Hirota–Maccari equation, are investigated by this means and new exact solutions are found.

Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method

A new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system. Compared with most of the existing tanh methods, the Jacobi elliptic function method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters. The solutions obtained in this paper include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions and (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper, KdV–MKdV, Ito's fifth MKdV, Hirota, Nizhnik–Novikov–Veselov, Broer–Kaup, generalized coupled Hirota–Satsuma, coupled Schrodinger–KdV, (2 + 1)-dimensional dispersive long wave, (2 + 1)-dimensional Davey–Stewartson equations. In addition, as an illustrative sample, the properties of the soliton solutions and Jacobi doubly periodic solutions for the Hirota equation are shown by some figures. The links among our proposed method, the tanh method, extended tanh method and the Jacobi elliptic function method are clarified generally.

A New Generalization of the (2+1)-dimensional Davey-Stewartson Equation

Abstract Using an asymptotically exact reduction method based on Fourier expansion and spatiotemporal re-scaling, a new integrable system of the nonlinear partial differential equation in (2+1)-dimensions, extended Davey-Stewartson I equation, is deduced from a known (2+1)-dimensional integrable equation. The integrability of the new equation system is explicitly proved by the spectral transformation. Actually, the corresponding Lax pair of the new equations can be obtained by applying the same reduction method to the Lax pair of the original equation.

The Davey - Stewartson equation and the Ablowitz - Ladik hierarchy

A relation between the Davey - Stewartson (DS) system and the Ablowitz - Ladik hierarchy (ALH) is established. It is shown that the former can be `embedded' into the system of flows described by the ALH. This relation is used to work out a procedure for obtaining a wide range of solutions of the (2 + 1)-dimensional DS equation using the traditional inverse scattering transform that has been developed for (1 + 1)-dimensional systems, without invoking the multidimensional one. The `embedding into the ALH' method is used to derive the dark soliton, quasiperiodic and some other solutions for the DS-II as well for the Ishimori equation, which is gauge equivalent to the DS system.

Bäcklund-Schlesinger transformations for Davey-Stewartson equations

It is shown that integrable (1+2)-dimensional Davey-Stewartson (DS) and Boiti-Leon-Pempinelli (BLP) equations admit an explicit invertible Backlund auto-transformation. An algorithm is developed to construct exact solutions for “flat-” and “horseshoe”-type solitons of the DS system. Successive application of these transformations allows us to find solutions of (1+1)- and (0+2)-dimensional Toda lattice equations. We point out a similar auto-transformation for analogues of the DS system realized for an arbitrary associative algebra with a unity, in particular, for matrix DS equations. We also relate the (1+2)-dimensional models that we construct to (1+1)-dimensional J-S-systems.