Generalized Davey Stewartson Research Articles

Theoretical Investigation on the Conservation Principles of an Extended Davey–Stewartson System with Riesz Space Fractional Derivatives of Different Orders

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications.

Applications of the R-Matrix Method in Integrable Systems

Based on work related to the R-matrix theory, we first abstract the Lax pairs proposed by Blaszak and Sergyeyev into a unified form. Then, a generalized zero-curvature equation expressed by the Poisson bracket is exhibited. As an application of this theory, a generalized (2+1)-dimensional integrable system is obtained, from which a resulting generalized Davey–Stewartson (DS) equation and a generalized Pavlov equation (gPe) are further obtained. Via the use of a nonisospectral zero-curvature-type equation, some (3+1) -dimensional integrable systems are produced. Next, we investigate the recursion operator of the gPe using an approach under the framework of the R-matrix theory. Furthermore, a type of solution for the resulting linearized equation of the gPe is produced by using its conserved densities. In addition, by applying a nonisospectral Lax pair, a (3+1)-dimensional integrable system is generated and reduced to a Boussinesq-type equation in which the recursion operators and the linearization are produced by using a Lie symmetry analysis; the resulting invertible mappings are presented as well. Finally, a Bäcklund transformation of the Boussinesq-type equation is constructed, which can be used to generate some exact solutions.

A new criterion and the limit of blowup solutions for a generalized Davey–Stewartson system

This paper concerns a generalized Davey–Stewartson system, which is one of the basic models in fluid mechanics. Applying a generalization of Lushnikov’s criteria, a new blowup criterion for a Davey–Stewartson system for the super mass-critical case is obtained. For super mass-critical and sub-cubic cases, we also derive the limit of blowup solutions at the blowup time, which mimics the dynamics of scattering 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.

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)].

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.

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.

On the integrability of a generalized Davey–Stewartson system

In this work we investigate the integrable cases of the elliptic–hyperbolic–hyperbolic generalized Davey–Stewartson system introduced in Babaoğlu and Erbay (2004) [6] following the method of Zakharov and Shulman (1980) [3]. This method provides us with a set of algebraic conditions on the parameters of the system, which are just necessary conditions for the system to be integrable by means of the inverse scattering transform. Taking into account the constraints arising from the physical derivation of the generalized Davey–Stewartson system as described in Babaoğlu and Erbay (2004) [6], we show that this system is integrable only when it can be transformed to an integrable case of the Davey–Stewartson system.

The Simplest Equation Method and Its Application for Solving the Nonlinear NLSE, KGZ, GDS, DS, and GZ Equations

A good idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the elliptic-like equations are derived using the simplest equation method and the modified simplest equation method, and then the exact solutions of a class of nonlinear evolution equations which can be converted to the elliptic-like equation using travelling wave reduction are obtained. For example, the perturbed nonlinear Schrödinger’s equation (NLSE), the Klein-Gordon-Zakharov (KGZ) system, the generalized Davey-Stewartson (GDS) equations, the Davey-Stewartson (DS) equations, and the generalized Zakharov (GZ) equations are investigated and the exact solutions are presented using this method.

On the blow-up phenomenon for a generalized Davey-Stewartson system

The blow-up solutions of the Cauchy problem for a generalized Davey–Stewartson system, which models the wave propagation in a bulk medium made of an elastic material with coupled stresses, are investigated. The mass concentration is established for all the blow-up solutions of the system. The profile of the minimal blow-up solutions as t → T (blow-up time) is discussed in detail in terms of the ground state.

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.

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.

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.

Self-similar solutions to a generalized Davey–Stewartson system

In this paper we study the global solvability and existence of self-similar solutions to a generalized Davey–Stewartson system. We first use the Banach fixed point theorem to establish a general global existence theorem. By using this general result to a class of special initial data we obtain the existence of global self-similar solutions.

A note on the global existence of small amplitude solutions to a generalized Davey–Stewartson system

In this paper, we are interested in the Cauchy problem for a generalized Davey–Stewartson (GDS) system. We establish the global time existence of small mass solutions for the GDS system in the elliptic–hyperbolic–hyperbolic case.

On the existence of special solutions of the generalized Davey–Stewartson system

In this note we show that for certain choice of parameters the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system admits time-dependent travelling wave solutions of the kind given in [V.A. Arkadiev, A.K. Pogrebkov, M.C. Polivanov, Inverse scattering transform method and soliton solutions for Davey–Stewartson II equation, Physica D 36 (1989) 189–197] for the hyperbolic Davey–Stewartson system. These solutions lead to radial solutions as well. We also find the sufficient conditions for non-existence of travelling wave solutions for the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system by using the point of view developed in [A. Eden, T.B. Gürel, E. Kuz, Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system, IMA J. Appl. Math. (in press)].

A cross-constrained variational problem for the generalized Davey–Stewartson system

AbstractWe study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson systemin ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering

We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(|x|^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$-solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudo-conformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type.We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.

Symmetry, full symmetry groups, and some exact solutions to a generalized Davey–Stewartson system

The Lie symmetry algebra of a generalized Davey–Stewartson (GDS) system is obtained. The general element of this algebra depends on eight arbitrary functions of time, which has a Kac–Moody–Virasoro loop algebra structure and is isomorphic to that of the standard integrable Davey–Stewartson equations under certain conditions imposed on parameters and arbitrary functions. Then based on the symmetry group direct method proposed by Lou and Ma [J. Phys. A 38, L129 (2005)] the full symmetry groups of the GDS system are obtained. From the full symmetry groups, both the Lie symmetry group and a group of discrete transformations can be obtained. Finally, some exact solutions involving sech-sech2-sech2 and tanh-tanh2-tanh2 type solitary wave solutions are presented by a generalized subequation expansion method.