Sokolov System Research Articles

Negative flows for several integrable models

A construction of negative flows for integrable systems based on the Lax representation and squared eigenfunctions is proposed. Examples considered include the Boussinesq equation and its reduction to the Sawada–Kotera and Kaup–Kupershmidt equations; one of the Drinfeld–Sokolov systems and its reduction to the Krichever–Novikov equation.

Persistence of Solitary Waves and Periodic Waves of a Singularly Perturbed Generalized Drinfel’d–Sokolov System

This paper focuses on the persistence of solitary waves and periodic waves of a singularly perturbed generalized Drinfel’d–Sokolov system. Geometric singular perturbation theory is first employed to reduce the higher-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we are then able to find some sufficient conditions about the wave speed to guarantee the existence of homoclinic orbits and periodic orbits, which indicates the existence of solitary waves and periodic waves. Furthermore, we find the lower and upper bounds of the limit wave speed.

CONSTRUCTION OF FRACTAL SOLITON SOLUTIONS FOR THE FRACTIONAL EVOLUTION EQUATIONS WITH CONFORMABLE DERIVATIVE

In this paper, the fractional evolutions are described by using the conformable derivative for the first time. We implement fractional functional variable method (FFVM) to obtain some new kinds of fractal soliton wave solutions for these fractional evolution equations. The simplicity and effectiveness of this proposed method are tested on the fractional Drinfeld–Sokolov system and fractional cubic Klein–Gordon equation. The FFVM provides a new perspective to construct exact fractal soliton wave solutions of complex fractional nonlinear evolution equations in mathematical physics.

An application of the exponential rational function method to exact solutions to the Drinfeld–Sokolov system

In the present study, we investigate solitary wave solutions to the Drinfeld–Sokolov system. This system describes the dispersive water waves in fluid mechanics and is known as an anisotropic integrable system along with a Lax extension of the famous Korteweg–De Vries equation. The pivotal aim of this study is to make much more contributions to the fluid mechanics by extracting various versions of traveling waves which are helpful to further understand the mechanism of propagation of the dispersive waves. The presented method used for achieving our main goal is the generalized exponential rational function method. Several different categories of solutions are all determined using a single structure in the utilized technique. After comparing with the existing methods, such as the Exp-function method, the generated exact wave solutions show the efficiency of the utilized method. For the sake of discovering more facts about the numerical nature of obtained wave solutions, several 2D diagrams have been shown. Further, the acquired results of this paper also pave the way for the use of the technique in solving other nonlinear models.

Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems

In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions.

The time fractional D(m,n) system: invariant analysis, explicit solution, conservation laws and optical soliton

ABSTRACT In this article, a generalized Drinfeld–Sokolov system, also called system which describes the anomalous propagation of nonlinear surface gravity waves over horizontal seabed, is investigated. The Lie classical method is performed on the governing model. The obtained symmetries are used to reduce the model into the fractional ordinary differential equations and to derive the local conservation laws. To make the use of the reduced system, an explicit series solution is derived. Also, it is shown that the bright and singular solitons exist only for the time fractional equation by means of a transformation. The considered system does not give rise to the dark soliton.

Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras

The article deals with the problem of the integrable discretization of the well-known Drinfeld–Sokolov hierarchies related to the Kac–Moody algebras. A class of discrete exponential systems connected with the Cartan matrices has been suggested earlier in Garifullin et al (2012 SIGMA 8 33) which coincide with the corresponding Drinfeld–Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been proven by numerous examples. In the present article we study those systems from this class which are related to the algebras . We found the Lax pair for arbitrary N, briefly discussed the possibility of using the method of formal diagonalization of Lax operators for describing a series of local conservation laws and illustrated the technique using the example of N = 3. Higher symmetries of the system are presented in both characteristic directions. The recursion operator for the case N = 3 is found. It is interesting to note that this operator is not weakly nonlocal.

On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems

This study investigates the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems by using the extended sinh-Gordon equation expansion method. The Nizhnik-Novikov-Veselov system is an isotropic Lax extension of the one-dimensional Korteweg-de Vries equation and the Drinfel’d-Sokolov system is a nonlinear model that possesses some special forms of Lax pairs. Topological kink-type, topological, non-topological, compound topological-non-topological bell-type and singular soliton solutions are successfully constructed. Under the choice of suitable values of parameters, the 2D, 3D and contour graphs to some of the obtained solutions are plotted.

Lie Point Symmetries and Travelling Wave Solutions for the Generalized Drinfeld–Sokolov System

ABSTRACTIn this article, we show successful methods with a wider applicability in physics, which have been applied to make an analysis of generalized Drinfeld–Sokolov system. Lie’s classical symmetries have been calculated and they have given us the Lie-group classification depending on the parameters. Moreover, for each case we have obtained their optimal system and reductions to conclude obtaining traveling wave solutions using the sine–cosine method.

Topological features of the Sokolov integrable case on the Lie algebra so(3,1)

The integrable Sokolov case on is investigated. This is a Hamiltonian system with two degrees of freedom, in which the Hamiltonian and the additional integral are homogeneous polynomials of degrees 2 and 4, respectively. It is an interesting feature of this system that connected components of common level surfaces of the Hamiltonian and the additional integral turn out to be noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed, and the topology of noncompact level surfaces is determined, that is, the closures of solutions of the Sokolov system on are described. Bibliography: 24 titles.

Topological features of the Sokolov integrable case on the Lie algebra

The integrable Sokolov case on is investigated. This is a Hamiltonian system with two degrees of freedom, in which the Hamiltonian and the additional integral are homogeneous polynomials of degrees 2 and 4, respectively. It is an interesting feature of this system that connected components of common level surfaces of the Hamiltonian and the additional integral turn out to be noncompact. The critical points of the moment map and their indices are found, the bifurcation diagram is constructed, and the topology of noncompact level surfaces is determined, that is, the closures of solutions of the Sokolov system on are described. Bibliography: 24 titles.

Compactons and topological solitons of the Drinfel'd–Sokolov system

This paper presents the Drinfel’d–Sokolov system (shortly D(m, n)) in a detailed fashion. The Jacobi’s elliptic function method is employed to extract the cnoidal and snoidal wave solutions. The compacton and solitary pattern solutions are also retrieved. The ansatz method is applied to extract the topological 1-soliton solutions of the D(m, n) with generalized evolution. There are a couple of constraint conditions that will fall out in order to exist the topological soliton solutions.

Systems of Kowalevski type and discriminantly separable polynomials

Starting from the notion of discriminantly separable polynomials of degree two in each of three variables, we construct a class of integrable dynamical systems. These systems can be integrated explicitly in genus two theta-functions in a procedure which is similar to the classical one for the Kowalevski top. The discriminantly separable polynomials play the role of the Kowalevski fundamental equation. Natural examples include the Sokolov systems and the Jurdjevic elasticae.

Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov systemut+(vm)x=0,vt+a(vn)xxx+buxv+cuvx=0calledD(m,n)system. We reveal some interesting bifurcation phenomena as follows. (1) ForD(2,1)system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) ForD(1,2)system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) ForD(2,2)system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

Auto-Bäcklund transformations and superposition formulas for solutions of Drinfeld–Sokolov systems

The paper is devoted to constructing Auto-Bäcklund transformations (ABT) and superposition formulas for the solutions of the Drinfeld–Sokolov (DS) systems. The transformations are derived from pairs of differential substitutions relating different systems of the DS type. The nonlinear superposition formulas for solutions of the DS systems are obtained from the assumption of commutativity of the Bianchi diagram. We indicate a seed solution for each system which can be used to generate multi-soliton solutions. As an application of the superposition formulas we construct two-soliton solutions for each of the DS systems.

Classification of single traveling wave solutions to the nonlinear dispersion Drinfel’d–Sokolov system

By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the nonlinear dispersion Drinfel’d–Sokolov system.

Traveling wave solutions [formula omitted] to a generalized Drinfel’d–Sokolov system which satisfy [formula omitted

An analysis of the coupled generalized Drinfel’d–Sokolov equationsut+α1uux+β1uxxx+γvδx=0andvt+α2uvx+β2vxxx=0is performed in the case of traveling wave solutions u and v satisfying the condition u=a1vm+a0. We are able to classify all such solutions in terms of the model parameters, and we then discuss the planar dynamics of such solutions. Numerical solutions are obtained and discussed for a variety of parameter regimes. Such results are one possible generalization of those given by Wu et al. [L. Wu, S. Chen, C. Pang, Traveling wave solutions for generalized Drinfeld–Sokolov equations, Applied Mathematical Modeling 33 (2009) 4126–4130].

Exp and modified Exp function methods for nonlinear Drinfeld–Sokolov system

In this paper, Exp-function and its modification methods have been applied to obtain an exact solution of the nonlinear Drinfeld–Sokolov system (DS). Modification of the method was first introduced by the same authors. The prominent merit of this method is to facilitate the process of solving systems of partial differential equations. These methods are straightforward and concise by themselves; moreover, their applications are promising to obtain exact solutions of various partial differential equations. It is shown that the methods, with the help of symbolic computation, provide very effective and powerful mathematical tools for solving such systems.

Bifurcations of travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov ( [formula omitted]) system

By using the bifurcation theory of planar dynamical systems to the nonlinear dispersion Drinfel’d–Sokolov ( D ( m , n ) ) system, the existence of solitary wave solutions, kink and anti-kink wave solutions, compacton solutions and uncountably infinite many smooth and non-smooth periodic wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.

Travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov ([formula omitted]) system

In this paper, travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov system (called D( m,n) system) are studied by using the Weierstrass elliptic function method. As a result, more new exact travelling wave solutions to the D( m,n) system are obtained including not only all the known solutions found by Xie and Yan but also other more general solutions for different parameters m,n. Moreover, it is also shown that the D( m,1) system with linear dispersion possess compacton and solitary pattern solutions. Besides that, it should be pointed out that the approach is direct and easily carried out without the aid of mathematical software if compared with other traditional methods. We believe that the method can be widely applied to other similar types of nonlinear partial differential equations (PDEs) or systems in mathematical physics.