DIFFERENTIAL INVARIANTS OF A GENERALIZED VARIABLE-COEFFICIENT GARDNER EQUATION

. In this paper, we consider a generalized variable-coeﬃcient Gardner equation. By using the equivalence group of this equation, we derive the diﬀerential invariants of ﬁrst order and the corresponding invariant equations. We employ these diﬀerential invariants and invariant equations to ﬁnd the most general subclass of variable-coeﬃcient Gardner equations which can be mapped into a speciﬁc constant-coeﬃcient equation by means of an equivalence transformation. Furthermore, diﬀerential invariants are applied to obtain exact solutions.


1.
Introduction. Over the last years, nonlinear partial differential equations (PDEs) with variable coefficients have become increasingly important because these describe many nonlinear phenomena more realistically than their constant coefficients counterparts. Among them, we highlight the Korteweg-de Vries (KdV) equation and its generalizations. The KdV equation is established as a simple mathematical model to describe the physics and the dynamics of shallow water. Moreover, it is an analytical model of tsunami generation by submarine landslides. The problem is that the KdV equation is quite a simple equation to analyze these phenomena. Therefore, generalizations of the KdV equation which involve more than one nonlinear term should be taken into account. The Gardner equation is one of them. The Gardner equation, also known as combined KdV-mKdV equation, is widely used in various areas of physics, such that plasma physics, fluid dynamics, quantum field theory, and it is a useful model for the description of a great variety of wave phenomena in plasma and solid state.
In recent years, several papers have been devoted to study the Gardner equation. In [11] and [13], Johnpillai and Khalique obtained the optimal system of one-dimensional subalgebras of the Lie symmetry algebras of the classes and where B(t) and Q(t) represent the dispersion term and the linear damping term respectively. Later, the same authors constructed conservation laws for equation (1) for some special forms of the functions B(t) and Q(t) [12]. Vaneeva et al. [34] considered the variable-coefficient Gardner equations where A(t), B(t) and C(t) are smooth functions satisfying BC = 0. Vaneeva et al. enhanced the classification of Lie symmetries obtained in [22] through the use of the general extended equivalence group. We propose herein a generalized variable-coefficient Gardner equation with nonlinear terms of any order where n is an arbitrary positive integer, A(t), B(t) = 0, C(t) and Q(t) are arbitrary smooth functions of t. The problem lies in the fact that the analysis of variable-coefficient equations seems rather difficult. Thus, one expects that there was a transformation which maps equation (3) into another equation of the same class with the least possible number of arbitrary functions. Equivalence transformations arise for deriving an comprehensive solution of the problem. Among its many applications, we highlight the fact that equivalence transformations can be used for classifying classes of differential equations with arbitrary elements [5,8,25,33]. Let us recall that an equivalence transformation is an invertible mapping of the dependent and independent variables of the form that maps every equation of class (3) into an equation of the same class which preserves the differential structure of the equation except maybe the form of the arbitrary functions,Ã(t),B(t),C(t) andQ(t). Furthermore, equivalence transformations have proved to be a powerful tool in the theory of differential invariants. In fact, deriving equivalence transformations is the first step towards the determination of differential invariants. A differential invariant is a real valued function in the augmented space of independent, dependent variables and its arbitrary elements, which is invariant under the group of equivalence transformations (4). A method for obtaining differential invariants of linear and nonlinear families of differential equations which admit an infinite-dimensional group of equivalence transformations was proved by Ibragimov in [9]. Differential invariants of several equations have been obtained making use of this method [2,3,28,31,32]. The linearization problem is a particular case of the equivalence problem when one of the equations is linear. This has been studied in detail in [10,29,30].
Differential invariants play an important role in the theory of differential equations. For instance, they are one of the essential pillars for constructing invariant differential equations and variational problems, and determining their explicit solutions and conservation laws. Moreover, the equivalence, symmetry and rigidity properties of submanifolds are all controlled by their differential invariants (see [23]). There are different methods to approach the equivalence problem, although it is worth stressing Lie's infinitesimal method. Lie proved that every invariant system of differential equations and every variational problem could be expressed by using differential invariants [17,18]. Moreover, Lie showed [18], how differential invariants can be used to integrate ordinary differential equations, and obtained a completed classification of all the differential invariants for all possible finite-dimensional Lie groups of point transformations. Based on Lie's results, Ovsyannikov [24] developed this approach for infinite Lie groups. For further information on the theory of differential invariants including algorithms of construction one can refer to [23,24].
The effort in finding exact solutions of nonlinear equations is crucial for understanding most nonlinear physical phenomena. There are several papers in which exact solutions of PDEs are obtained from the similarity reductions [1,4,7,26]. The great utility of similarity solutions is that they may be calculated by solving an ordinary differential equation (ODE) instead of a PDE. Nevertheless, it is not always obvious how to integrate these ODEs. An alternative approach is to consider another equation, more simple to study or with a known solution, which can be mapped into the equation under consideration through an invertible transformation. For a family of PDEs, differential invariants deal with this equivalence problem.
The structure of the paper is the following: In Section 2 we present the infinitesimal generators of the equivalence group E of equation (3). In Section 3 we derive the differential invariants of first order and its invariant equations with respect to the equivalence group. In Section 4 we characterize the subclasses of equation (3) which can be mapped into a specific constant-coefficient equation. Finally, as an application of the differential invariants obtained, we determine exact solutions for special subclasses of equation (3). (3) is a nondegenerate point transformation from (t, x, u) to (t,x,ũ) with the property that it preserves the differential structure of the equation, that is, it transforms any equation of class (3) to another equation of the same class except its arbitrary functions,Ã(t),B(t),C(t) andQ(t). The set of equivalence transformations forms a group denoted by E. We search for the equivalence operator Y given by

Equivalence transformations. An equivalence transformation of class
where τ , ξ and η depend on t, x and u. On the other hand, the functions w i , which correspond to the functions A, B, C and Q, depend on t, x, u, A, B, C and Q. In [6], it was proved that class (3) admits an infinite-dimensional equivalence group E which is spanned by the following generators where α = α(t) and r = r(t) are arbitrary smooth functions. From generators (6)-(9) the finite form of the equivalence transformations was obtained where 1 , 2 and µ are arbitrary constants, λ = λ(t) and µ = µ(t) are arbitrary smooth functions with λ t = 0.

Differential invariants of class (3).
Our aim is to obtain the differential invariants for class (3). We found that class (3) admits an infinite-dimensional continuous group E spanned by (6)- (9). We call a nth-order differential invariant of class (3) to a real valued function J which is invariant under the equivalence group E extended to the derivatives of A, B, C and Q up to the nth order. In the case that derivatives of the arbitrary elements do not appear, we call it differential invariants of order zero. Furthemore, any equation is the sth prolongation of the operator Y k , is said to be an invariant equation.
3.1. Differential invariants of zero order. To begin with, we search for differential invariants of zero order, J(t, x, u, A, B, C, Q). Due to the fact that in class (3) the arbitrary elements only depend on t, we seek for invariants of the form In order to get the differential invariants of zero order we must require that (11) is invariant under the equivalence group E. This leads to a system of linear PDEs of first order in J given by Y k (J) = 0, k = 1, 2, α, r.
Solving this system we get the expected differential invariants of zero order. Invariant test Y 2 (J) = 0 is trivially satisfied. The remaining invariant tests lead us to J = constant. Therefore, class (3) does not admit differential invariants of zero order.
3.2. Differential invariants of first order. At this point, we should search the first order invariants of class (3). To achieve this objective, we need to determine the first prolongation of the operator Y (5): . . , 4, represents each function (A, B, C, Q). The operatorD t is defined as For further information on how higher order prolongations can be constructed, one can refer, as example, to references [27,28]. The invariant test applied to the function J given by

DIFFERENTIAL INVARIANTS GARDNER EQUATION 751
The first prolongations of the generators Y r , keeping only the elements we need are given by Since α(t) and r(t) are arbitrary functions, the invariance criterion (13) leads to a system of 7 linear first order PDEs whose independent solutions provide the set of differential invariants of first order. This system yields two differential invariants of first order are invariant equations.
4. Applications of differential invariants. Theory of differential invariants deals with different problems. First, one can determine if two given PDEs are equivalent with respect to an invertible transformation. Once established the equivalence, we are interested in obtaining a transformation which connects these equivalent equations. In that case, it is possible to find exact solutions of a PDE from the solutions of equations which are equivalent to this one. In order to derive equivalent equations we will make use of the following result: two equations are equivalent with respect to an equivalence transformation iff all their invariants are equal and they satisfy the same invariants equations. According to (17), since A = 0 is an invariant equation, any equation of class (3) with A = 0 cannot be equivalent to an equation of the same class with A = 0 by an equivalence transformation. Likewise, for B = 0 and C = 0. Let us remember we have already supposed that B = 0. Therefore, we will focus our attention on equations of class (3) satisfying ABC = 0.
Our motivation is to transform an equation belonging to class (3) to a constantcoefficient equation through equivalence transformations. Taking into account the results obtained in Sections 2 and 3, we use differential invariants and invariant equations to determine all equations of class (3) which can be mapped under an equivalence transformation of the form (10) into the constant-coefficient equation where m 1 , m 2 , m 3 are arbitrary nonzero constants. For equations of subclass (18) is Equations of subclass (18) satisfy the invariant equations (17). Thus, if we solve the invariant equations (17), we will obtain neccesary conditions for an equation to be mapped into (18). From (17), we deduce that the more general form of equations of class (3) which are equivalent to (18) is where c 1 , c 2 are arbitrary nonzero constants, G(t) = 0 and H(t) are arbitrary functions. It can be easily checked that the transformation maps equation (18)  We can look for travelling wave solutions for equation (18). Equation (18) with n = 1 was studied by several authors [14,19,20,21]. This equation is integrable once by the inverse scattering transform, it has infinitely many conservations laws and many rational and solitary wave solutions. Lie symmetries, Lax pair, rational solutions, soliton solutions and other remarkable properties for this equation was obtained in [15].
Taking into account the travelling wave we obtain from equation (18) the nonlinear ODE Integrating (23) with respect to z we get where k 1 is an integrating constant. Multiplying equation (24) by h and integrating once with respect to z we get the nonlinear equation
For the sake of simplicity, we will only take a case for n = 1 and another for n = 2. From F 1 (z) in Table 1 with = 1 we obtain the following solution of equation (23) for n = 1: Undoing transformation (22) we obtain a solution of equation (18) u(t, x) = 6λ Finally, by using transformation (20), is a solution of equation (21) for n = 1. Similarly, from F 1 (z) in Table 2  , is a solution of equation (21) for n = 2. Here λ m 2 > 0 and f t ,x is given by (33).

5.
Conclusions. In this work, a generalized variable-coefficient Gardner equation has been considered. Through the use of an infinitesimal technique based on the equivalence group we have obtained differential invariants along with their invariants equations. We have determined that equation (3) does not admit differential invariants of zero order although it admits differential invariants of first order. Based on the fact that two equations are equivalent with respect to an equivalence transformation iff all their invariants are equal and they satisfy the same invariants equations, we have obtained the most general subclass (19) of equation (3) which can be mapped into a specific constant-coefficient equation (18) by an equivalence transformation. Moreover, we have determined the explicit form of this transformation. Then, we have shown exact solutions of equation (18). Finally, by using transformation (20) we construct exact solutions of the wide equivalent class of equations (19).