Double reductions/analysis of the Drinfeld–Sokolov–Wilson equation

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Abstract

In this paper, we show that the recently developed formulation of the association between symmetries and conservation laws lead to double reductions of the third-order Drinfeld–Sokolov–Wilson system of partial differential equations (PDEs). For illustrative purposes only, we carry out the procedure for a version of the well-known third-order scalar Korteweg–de Vries (KdV) equation via a scaling symmetry of the equation.

Introduction

The theory of double reduction of a PDE (or systems of PDEs) is well-known for the association of conservation laws with Noether symmetries [1], [11]. The association of conservation laws with Lie–Bäcklund symmetries [7] and non-local symmetries [12], [13], [14] was analysed. This lead to the expansion of the theory of double reduction for PDEs with two independent variables which do not possess a Lagrangian formulation, i.e., do not possess Noether symmetries [15]. In this article, we apply the fundamental theorem of double reduction for a third-order system of PDEs with two independent variables, where the Lie point symmetry is associated with the conservation law.

The Drinfeld–Sokolov–Wilson (DSW) system of equations has been analysed in many texts, e.g., [3], [11], [16]. The system possesses an infinite number of conservation laws and has a Lax representation. It presents interesting and different types of solutions including static solitons; these are static solutions that interact with moving solitons without deformations [4]. Most of the previous analyses are based on the ‘travelling wave’ type solutions via some well known substitutions. There are also a number of numerical approaches to the problem. The method adopted here, firstly, shows that the travelling wave method, by the underlying symmetries of the equation, is recovered and, more importantly, the solutions are obtained via a double reduction following an association of the symmetry with conserved vectors of the equation. Such an association may and, indeed does, exist for a range of symmetries, not only with those related to the travelling wave ones. This will be demonstrated with the scaling symmetries of the equation which lead to new exact solutions. Also, there exists additional conserved forms of the DSW equation for special cases of the parameters [10] and, therefore, possibilities for additional solutions may exist. We will perform reductions based on this property of the special cases. After a double reduction, we use Mathematica to present a numerical simulation for one of the cases.

Finally, it is worth noting that in [10], the multiplier approach was used to obtain local conserved vectors and in [9], the relationship between the Lie point symmetry generators and the multipliers of this system of PDEs was investigated.

This system is given by the equationsut+2vvx=0,vt-avxxx+3buxv+3kuvx=0.

The well-known scalar KdV equation has been extensively analysed through many approaches. Merely as an illustrative example, we first perform the procedure of double reduction for this equation. Since the travelling wave solution is well-known, we do not perform the double reduction for this case. Instead, we consider a reduction via a scaling symmetry with some conserved vector. The illustration is performed on the version of the KdV given byut-uxxx-uux=0.We refer the reader to the references [2], [6], [7], [8] for the preliminaries.

Section snippets

Illustrative example: double reduction of the scalar KdV equation

We analyse the following scalar PDEG=ut-uxxx-uux=0.Eq. (2.1) admits the following four Lie point symmetriesX1=t,X2=tt+13xx-23uu,X3=-tx+u,X4=xand the following three conserved vectorsT1=12u2,-uuxx+12ux2-13u3,T2=12tu2+xu,-tuuxx-xuxx+12tux2+ux-13tu3-12xu2,T3=u,-uxx-12u2,with the corresponding multipliersq1=u,q2=x+tu,q3=1.

Double reductions of the Drinfeld–Sokolov–Wilson equation

In this section, we analyse the following system of PDEsG1=ut+2vvx=0,G2=vt-avxxx+3buxv+3kuvx=0.Eq. (3.1) admits a three-dimensional Lie point symmetry algebra spanned byX1=t,X2=x,X3=3tt+xx-2uu-2vv.

Conclusion

We have shown how the interplay between the underlying symmetries and conservation laws of a system of PDEs lead to reductions (double) of a class of Drinfeld–Sokolov–Wilson equations. In all of the three cases on the specific relationship of the parameters b and k, we obtained a reduction to an ODE of order, at most, two. After performing the double reduction procedure for one of the cases, we adopted a numerical approach via Mathematica to illustrate the profile of the solution for one of the

Acknowledgements

The authors thank the editor and all the reviewers for their list of corrections, comments and suggestions.

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