Group Analysis and Nonlinear Self-Adjointness for a Generalized Breaking Soliton Equation

https://doi.org/10.1016/S0034-4877(15)60026-XGet rights and content

In this paper, a variable coefficient (2 + 1)-dimensional generalized breaking soliton equation is considered by means of the Lie group method. Having written an equation as a system of two dependent variables, we perform a complete group classification for the system. Consequently, for arbitrary functions, solutions of the generalized breaking soliton equation are connected with the ones of a variable coefficient Korteweg–de Vries equation by a transformation. For other cases, the reduced equations and exact solutions are constructed. Meanwhile, we prove that the system is nonlinearly self-adjoint and construct the general conservation law formulae.

Introduction

It is well known that modeling the phenomena in nature by partial differential equations (PDEs) is one of the central problems of mathematical physics and applied mathematics and thus attracts attention of researchers in the associated fields. In order to obtain more accurate information about the models, some PDEs contain some arbitrary parameters or functions which are not fixed and later must be determined by practical applications, also some PDEs have been extended to higher dimensions [1]. For instance, the Kadomtsev–Petviashvili equation, which has two-periodic wave solutions characterizing the shallow water waves with more accuracy, is a (2 + 1)-dimensional extension of the celebrated Korteweg–de Vries (KdV) equation [2].

The problem of study of higher-dimensional PDEs involving arbitrary parameters or functions, such as constructing exact solutions and conserwation laws, triggered more new methods. The theory of Lie groups and Lie algebras, introduced by Sophus Lie (1842–1899) and also named symmetry group, evolved into one of the most important development of mathematics and physics and exerted important effects in diverse fields [3, 4]. For example, the symmetry group admitted by PDEs can transform higher-dimensional PDEs to the familiar lower-dimensional ones and then allows to obtain exact solutions of the original PDEs. It can also help to determine arbitrary elements in the PDEs to search for further information about the models.

Symmetry also plays an important role in constructing conservation laws of PDEs. When PDEs admit a variational structure, Noether theorem gives the general conservation law formula by means of variational symmetries. If PDEs are not obtained from a variational principle, some new methods are developed to achieve the goal [5, 6, 7]. Recently, the concept of nonlinear self-adjointness [7] including the subclasses stated earlier [8, 9], provided a feasible method to construct conservation laws of PDEs. The main idea of the method, which traces back to the literature [10, 11] and references therein, is to turn the system of PDEs into Lagrangian equations by artificially adding additional variables, then to apply the theorem proved in [12] to construct conservation laws. Approximate nonlinear self-adjointness and approximate conservation laws were considered in [13].

Since many useful models in theoretical and applied sciences admit rich symmetry structures that follow from physical laws, such as e.g. from Galilean or relativistic theories, thus another problem is to determine which differential equation selecting from a broad class of possible PDEs is the best model to reflect the natural laws from the point of view of the Lie group theory [14]. The answer to this question is the group classification. The main idea of group classification is to classify the arbitrary parameters or functions contained in PDEs which make that PDEs admit more symmetry groups. Generally speaking, to find complete group classifications of such equations in terms of their unknown parameters is a complicated problem that challenges researchers. The principal tool for handling group classification is the classical infinitesimal routine developed by Sophus Lie. It transforms the problem to finding the corresponding Lie symmetry algebra of infinitesimal operators whose coefficients are found as solutions of some over-determined system of linear PDEs [3, 4].

Here, we will pay attention to a variable-coefficient (2+1)-dimensional generalized breaking soliton equation of the form [15] ut+a(t)uxxx+6a(t)uux+b(t)uxxy+4b(t)(uuydx)x=0,where x and y are the scaled space coordinates, t is the scaled time coordinate, u is a function of x, y and t, and a(t) and b(t) are analytical functions of t. Eq. (1) reduces to the variable coefficient KdV equation when y = x or b(t) = 0. Bilinear forms and N-soliton solutions of Eq. (1) are obtained in [16]. For Eq. (1), we assume that at least one of a(t) and b(t) ≠ 0 is a nonconstant function because the constant case had been studied in [17, 18, 19, 20, 21, 22].

Historically, there are a number of papers contributing to the studies of subclasses of Eq. (1). For example, a ‘typical’ (2+1)-dimensional breaking soliton equation ut+ωuxxy+4ω(uuydx)x=0is a particular case of Eq. (1) with a(t) = 0 and b(t) = ω (constant). Various solutions have been reported in [17, 18, 19, 20] and references therein.

When a(t) = a, b(t) = b, where a, b are constants, Eq. (1) is written as ut+auxxx+6auux+buxxy+4b(uuydx)x=0,which describes a (2 + 1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis; it has been studied from the point of view of the Painlevé property, dromion-like structures and Wronskian solutions, etc. [21, 22].

From the above statements, it is easy to see that Eq. (1) is a generalization of many physically important systems, thus there is an essential interest in investigating it explicitly from the group theoretical point of view. The goal of this paper is to present a complete group analysis and to study nonlinear self-adjointness for Eq. (1). Some new interesting extensions of the Lie symmetry group and conservation laws have been obtained for these equations.

Using the transformation t˜=a(t)dt,x˜=x,u˜=u, Eq. (1) is transformed to ut+uxxx+6uux+b(t)a(t)[uxxy+4(uuydx)x]=0,after deleting the symbol ‘’, then the two arbitrary functions a(t) and b(t) can be combined into a single new function. Nevertheless, here we prefer to deal with Eq. (1) because several models can easily be constructed and classified in terms of a(t) and b(t) and also some cases are avoided such as Eq. (2).

After introducing transformation v = ∫ uy dx, Eq. (1) can be written as a system of two coupled PDEs F1=vxuy=0,F2=ut+a(t)uxxx+6a(t)uux+b(t)uxxy+4b(t)(uv)x=0.

In particular, the connection of solutions between Eq. (1) and Eqs. (5) is stated as follows. If (u, v) = (U(x, t), V(x, t)) solves Eqs. (5), then u = U(x, t) solves Eq. (1). Conversely, for any u = U(x, t) solving Eq. (1), there exists a pair of functions (u, v) = (U(x, t), V(x, t)) solving Eqs. (5) with V(x, t) unique to a constant. Thus in what follows, we will perform group analysis and nonlinear self-adjointness for Eqs. (5) instead of Eq. (1).

The rest of the paper is outlined as follows. In Section 2, we perform group analysis for Eqs. (5). Complete group classifications are given and the corresponding symmetry reductions are constructed. Section 3 concentrates on the nonlinear self-adjointness and the construction of conservation laws for Eqs. (5). The last section summarizes the results of the paper.

Section snippets

Group analysis

In this section, we first perform group classification for Eqs. (5), and then use it to reduce Eqs. (5) to (1 + 1)-dimensional PDEs.

Nonlinear self-adjointness

In this section we describe nonlinear self-adjointness and construct conservation laws for Eqs. (5).

Summary

In this paper, after writing Eq. (1) as the equivalent system (5), we performed a complete group classification of Eqs. (5). The reduced equations and exact solutions have been constructed. Moreover, we showed that Eqs. (5) is nonlinearly self-adjoint and the general conservation law formulae have been constructed.

Acknowledgements

This paper is supported in part by the NSF of China No. 11301012 and the Doctoral Fund of North China University of Technology. The work of Chen is supported in part by the NSF of China No. 11271363.

References (24)

  • A.H. Kara et al.

    Noether-type symmetries and conservation laws via partial Lagrangians

    Nonlinear Dyn.

    (2006)
  • N.H. Ibragimov

    Nonlinear self-adjointness in constructing conservation laws

    Arch. ALGA

    (2011)
  • Cited by (1)

    • Adjoint symmetry and conservation law of nonlinear diffusion equations with convection and source terms

      2016, Nonlinear Analysis: Real World Applications
      Citation Excerpt :

      For the PDEs without having a variational principle, one may adopt direct method, partial Lagrangian method, multiplier method, nonlinear self-adjointness method to achieve the goal [8–12]. In particular, nonlinear self-adjointness provides an effective method to construct conservation laws of the system of PDEs whether it has a variational principle or not [13–15]. The main idea of the method is to turn the system of PDEs into Lagrangian equations by artificially adding additional variables, then to apply the theorem proved in [16] to construct local and nonlocal conservation laws.

    View full text