Self-adjointness, Group Classification and Conservation Laws of an Extended Camassa-Holm Equation

Copyright: © 2015 Nadjafikhah M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Abstract In this paper, we prove that equation 2 2 3 ( ) = 0 t x x x t x x E f a b ≡ − + − − u u u u u u uu is self-adjoint and quasi self-adjoint, then we construct conservation laws for this equation using its symmetries. We investigate a symmetry classification of this nonlinear third order partial differential equation, where f is smooth function on u and a, b are arbitrary constans. We find Three special cases of this equation, using the Lie group method.


Introduction
A new procedure for constructing conservation laws was developed by Ibragimov [1]. For Camassa-Holm equation are calculated in studies of Ibragimov, Khamitova and Valenti [2]. In this paper, we study the following third-order nonlinear equation 2 2 and we show that this equation is self-adjoint and quasi self-adjont. Therefore we find Lie symmetries and conservation laws. There are three cases to consider: 1) b ≠ 0, a = arbitrary constant, 2) b = 0, a ≠ 0, and 3) b = 0, a = 0. Clarkson, Mansfield and Priestly [3] are concerned with symmetry reductions of the non-linear third order partial differential equation given by , where ∈, k, and β are arbitrary constants. Symmetry classification and conservation laws for higher order Camassa-Holm equation are calculated in framework of Nadjafikhah and Shirvani-Sh [4].

Preliminaries
In this section, we recall the procedure in literature of Ibragimov [1]. Let us introduce the formal Lagrangian where v = v(t, x) is a new dependent variable.
We define the adjoint equation by is the variational derivative and D i is the operator of total diferentiation.
An equation E = 0 is said to be self-adjoint [5] if the equation obtained from the adjoint equation by substitution v = u is identical with the original equation.
An equation E = 0 is said to be quasi-self-adjoint [5] if there exists a function = ( ) with an undetermined coefficient λ. Eq.(1) is said to have a nonlocal conservation law if there exits a vector C = (C 1 , C 2 ) satisfying the equation on any solution of the system of differential equations comprising (E) and the adjoint equation (E * ). We say that orginal equation has a local conservation law if (3) is satisfied on any solution of Eq.(1). In studies of Ibragimov [1], the conserved vector associated with the Lie point symmetry 1 2 = ( , , ) ( , , ) ( , , ) is obtained by the following formula : where i, j, k = 1,2 and = We recall the general procedure for determining symmetries for an arbitrary system of partial differential equations [6]. Let us consider the general system of a nonlinear system of partial differential equations of order n, containing p independent and q dependent variables is given as follows and the derivatives of u with respect to x up to n, where u (n) represents all the derivatives of u of all orders from 0 to n. We consider a one-parameter Lie group of transformations acting on the variables of system (5): , where i=1,⋅⋅⋅, p, j = 1,⋅⋅⋅, q. ξ i , φ j are the infinitesimal of the transformations for the independent and dependent variables, respectively, and ∈ is the transformation parameter. We consider the general vector field v as the infinitesimal generator associated with the above group A symmetry of a differential equation is a transformation, which maps solutions of the equation to other solutions. The invariance of the system (5) under the infinitesimal transformation leads to the invariance conditions. (Theorem 2.36 of studies of Olver [6], Theorem 6.5 of literature of Olver [7]).
where v n is called the n th order prolongation of the infinitesimal generator given by

Adjoint Equation and Classical Symmetry Method
Formal Lagrangian for Eq. (1) is Therefore, the adjoint equation E * to Eq. (1) is Hence, Eq. (1) is self-adjoint if and only if it has the form a = 2b.
Consider again Eq. (1), and substitute in the adjoint equation (8), then Hence, Eq. (1) is quasi self-adjoint if and only if it has the form In this section, we will perfom Lie group method for Eq. (1) on (x 1 = x, x 2 = t, u 1 = u), where ε ≤ 1 the group parameter and 1 = ξ ξ , 2 = ξ τ and 1 = φ φ are the infinitesimals of the transformations for the independent and dependent variables respectively. The associated vector fields is of the form = ( , , ) and the third porolongation of v is the vector field where D k is the total derivative with respect to independent variables. The invariance condition (6) for Eq. (1) is given by, whenever E = 0. The condition (12) is equvalent to whenever E = 0. Substituting (11) into (13), yields the determining equations. There are three cases to consider:
x t u t x We want to construct the conservation law associated with the symmetry 1 ( 1) = .  The right-hand side of (4) is written x t x We eliminate the term ξ i L since the Lagrangian L is equal to zero on solution of Eq.(1). Substituting in (29), the expression (7) for L and (28) for W, we obtain We can eliminate u t by using Eq.(1) and then substitute in (30) and (31) the expression v = u, therefore arrive at the conserved vector with the following components:  = , Now, by considering Eq. (33) -(42) it is not to hard to find that the components ξ, τ and φ of infinitesimal generators become To find complete solution of the above system, we consider Eq. (43) and l = dim Spam  {f u , f,1}. Three general cases are possible: 3.2.i) l = 1, then f = constant; 3.2.ii) l = 2, then f u = αf + β ; 3.3.iii) l = 3, then αf u + βf + γ ≠ 0, α ≠ 0.