Generalized Davey-Stewartson Equations Research Articles

The exact solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities using the dynamical system and the first integral methods

Abstract The exact traveling wave solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities are studied using the dynamical system and the first integral methods. Taking different parameter conditions, we obtain periodic wave solutions, exact solitary wave solutions, kink wave solutions, and anti-kink wave solutions.

New and more fractional soliton solutions related to generalized Davey–Stewartson equation using oblique wave transformation

The generalized fractional Davey–Stewartson (DSS) equation with fractional temporal derivative, which is used to explore the trends of wave propagation in water of finite depth under the effects of gravity force and surface tension, is considered in this paper. The paper addresses the full nonlinearity of the proposed model. To extract the oblique soliton solutions of the generalized fractional DSS (FDSS) equation is the dominant feature of this research. The conformable fractional derivative is used for fractional temporal derivative and oblique wave transformation is used for converting the proposed model into ordinary differential equation. Two state-of-the-art integration schemes, modified auxiliary equation (MAE) and generalized projective Riccati equations (GPREs) method have been employed for obtaining the desired oblique soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, and periodic solitary wave solutions. The occurrence of these results ensured by the limitations utilized is also exceptionally promising to additionally investigate the propagation of waves of finite depth. The latest found solutions with their existence criteria are considered. The 2D and 3D portraits are also shown for some of the reported solutions. From the graphical representations, it have been illustrated that the descriptions of waves are changed along with the change in fractional and obliqueness parameters.

Soliton solutions of the generalized Davey-Stewartson equation with full nonlinearities via three integrating schemes

This paper considers the generalized Davey-Stewartson equation that is used to investigate the dynamics of wave propagation in water of finite depth under the effects of gravity force and surface tension. The model is considered in the presence of full nonlinearity. The main objective of this paper is to extract soliton solutions of the generalized Davey-Stewartson equation. Three state-of-the-art integration schemes, namely exp(-Φ(ξ))-expansion method, the first integral method and the Sine-Gordon expansion method have been employed for obtaining the desired soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, rational and periodic solitary wave solutions. All the newly found solutions are discussed with their existence criteria. The 2D and 3D portraits are also shown for some of the reported solutions.

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

New Soliton Solutions of the (2+1)-Dimensional System Davey-Stewartson Equation

In this paper, we derive several soliton solutions of the generalized Davey-Stewartson equation with the complex coefficients. First we use the travelling wave transformation to reduce the initial system to ODE. The equivalent ODE is then solved, giving several classes of solutions, depending on the values of the parameters. Finally, the Extended Tanh-Coth method and Modified simple equation method.

Exact traveling wave solutions of the generalized Davey-Stewartson equation

Exact traveling wave solutions of the generalized Davey-Stewartson equation

New Exact Solutions of the Davey–Stewartson Equation with Power-Law Nonlinearity

This work obtains the soliton solutions of the generalized Davey–Stewartson equation with the complex coefficients. First, the extended Weierstrass transformation method is used to carry out the solutions of this equation, and some new solutions, known as Weierstrass elliptic function solutions, are obtained by this method. Then, the trial equation method is used to obtain the soliton solutions of this equation.

Lie-algebraic symmetries of generalized Davey-Stewartson equations

We identify the full Lie-algebraic structure of the generalized Davey-Stewartson (GDS) system of equations with symmetries of a specific of continual Lie algebras. In particular, we show that they are related to two copies of the Poisson bracket continual Lie algebra.

Exact solutions of the ZK-MEWequation and the Davey-Stewartson equation

In this paper we introduce a new version of the trial equation method for solving non-integrable partial difierential equations in mathematical physics. Some exact solutions including soliton solutions, rational and elliptic function solutions to the generalized (2+1)-dimensional ZK-MEW equation and the generalized Davey-Stewartson equation with the complex coe‐cients are obtained by this method.

A note on the Cauchy problem of the generalized Davey–Stewartson equations

In this note, we establish some local and global existence results for the Cauchy problem of a class of nonlinear dispersive equations which generalize the nonlinear Schrödinger equations and the Davey–Stewartson equations. These results improve some previously obtained results by some other authors when they are restricted to certain special equations.

Long-wave short-wave resonance case for a generalized Davey–Stewartson system

It is observed that the generalized Davey–Stewartson equations are not valid for a long-wave short-wave resonance case. In the case where the phase velocity of the long longitudinal wave is equal to the group velocity of the short transverse wave, new ( 2 + 1 ) dimensional evolution equations, called the long-wave short-wave interaction equations, are derived to describe the resonance case. The special solutions of the long-wave short-wave interaction equations are also obtained in terms of Jacobian elliptic functions.

The generalized Davey-Stewartson equations, its Kac-Moody-Virasoro symmetry algebra and relation to Davey-Stewartson equations

We compute the Lie symmetry algebra of the generalized Davey-Stewartson (GDS) equations and show that under certain conditions imposed on parameters in the system it is infinite-dimensional and isomorphic to that of the standard integrable Davey-Stewartson equations which is known to have a very specific Kac-Moody-Virasoro loop algebra structure. We discuss how the Virasoro part of this symmetry algebra can be used to construct new solutions, which are of vital importance in demonstrating existence of blow-up profiles, from known ones using Lie subgroup of transformations generated by three-dimensional subalgebras, namely sl(2,R). We further discuss integrability aspects of GDS equations.

Some special solutions of a generalized Davey–Stewartson system

Abstract A system of three equations which is called generalized Davey–Stewartson equations is considered. Some special solutions of these equations are investigated by both a tanh method and a variable separation approach.

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.

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.

Energy Scattering for the Generalized Davey-Stewartson Equations

Considering the generalized Davey-Stewartson equation $$i\mathop u\limits^. - \Delta u + \lambda \left| u \right|^p u + \mu E\left( {\left| u \right|^q } \right)\left| u \right|^{q - 2} u = 0$$ where $$\lambda > 0,\mu \ge 0,E ={\mathcal {F}}^{ - 1} \left( {\xi _1^2 /\left| \xi \right|^2 } \right){\mathcal{F}}$$ we obtain the existence of scattering operator in ∑(ℝ n ) := { u ∈ H 1(ℝ n ) : |x|u ∈ L 2(ℝ n )}.

Darboux Covariant 2+1 Dimensional Soliton Equations of Inhomogeneous Type

We present 2+1 dimensional soliton equations of inhomogeneous type which are derived systematically using the covariance with respect to the Darboux transformation. By setting Darboux invariants to be explicit functions of space-time, we obtain nonlinear equations describing soliton propagation in media with inhomogeneities. Darboux covariant equations obtained here include the inhomogeneously generalized Davey-Stewartson equation, the 2+1 dimensional modified Korteweg-de Vries equation and other binary Darboux covariant equations of inhoinogeneous type.

Nonlinear modulation of multidimensional lattice waves.

The equations governing weakly nonlinear modulations of N-dimensional lattices are considered using a quasidiscrete multiple-scale approach. It is found that the evolution of a short wave packet for a lattice system with cubic and quartic interatomic potentials is governed by the generalized Davey-Stewartson (GDS) equations, which include mean motion induced by the oscillatory wave packet through cubic interatomic interaction. The GDS equations derived here are more general than those known in the theory of water waves because of the anisotropy inherent in lattices. The generalized Kadomtsev-Petviashvili equations describing the evolution of long-wavelength acoustic modes in two- and three-dimensional lattices are also presented. Then the modulational instability of an N-dimensional Stokes lattice wave is discussed based on the N-dimensional GDS equations obtained. Finally, the one- and two-soliton solutions of two-dimensional GDS equations are provided by means of Hirota's bilinear transformation method.

Line soliton solutions for a generalized Davey-Stewartson equation with variable coefficients

The Darboux transformation for a generalized Davey-Stewartson equation is constructed. Expressions for line solitons are obtained explicitly by one and two applications of Darboux transformations.

Generalized Davey-Stewartson equations associated with symmetric spaces

Generalized Davey-Stewartson equations associated with symmetric spaces