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**Nadjafikhah M ^{1*} and Pourrostami N^{2}**

^{1}Department of Pure Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak-16, Tehran, Iran

^{2}Department of Complementary Education, Payam Noor University, Tehran, Iran

- Corresponding Author:
- Nadjafikhah M

Department of Pure Mathematics School

of Mathematics Iran Universityof Science and

Technology, Narmak-16, Tehran, Iran+98 2173225426

Tel:+982173228426

Fax:[email protected]

E-mail:

**Received date:** April 21, 2015; **Accepted date:** December 22, 2015; **Published date:** December 24, 2015

**Citation:** Nadjafikhah M, Pourrostami N (2015) Self-adjointness, Group Classification and Conservation Laws of an Extended Camassa-Holm Equation. J Generalized Lie Theory Appl S2:004. doi:10.4172/1736- 4337.S2-004

**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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

In this paper, we prove that equation 2 ( ) 2 3 = 0 t x t x x x x E ≡ u −u + u f u − au u − buu 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.

**Lie symmetry **analysis; Self-adjoint; Quasi **self-adjoint**; **Conservation laws**; Camassa-Holm equation; Degas peris-Procesi equation; **Fornberg whitham equation**; BBM equation

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

(1)

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].

The special cases of (1) are:

Camassa-Holm (CH) equation k-arbitrary (real), describing the unidirectional propagation of shallow water waves over a flat bottom (let f = k + 3u, a = 2, b = 1 in (1).

Degas peris-Procesi (DP) equation k-arbitrary (real), is another equation of this class (let f = k + 4u, a = 3, b = 1 in (1).

Fornberg Whitham (FW) equation is another equation of this class (let f = 1 + u, a = 3, b = 1 in (1)).

**BBM equation ** , is another equation of this class (let f = 1 + u, a = 0, b = 0 in (1)).

In this section, we recall the procedure in literature of Ibragimov [1]. Let us introduce the formal Lagrangian

L ≡ vE, (2)

where *v = v(t, x) *is a new dependent variable.

We define the adjoint equation by Here

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 such that 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

(3)

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 is obtained by the following formula :

(4)

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

(5)

involving and the derivatives of u with respect to x up to n, where 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 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]).

(6)

where v_{n} is called the nth order prolongation of the infinitesimal generator given by where and the sum is over all k’s of order 0 < #k ≤ n. If #k = α, the coeficent will depend only on α’th and lower order derivatives of u and where

Formal Lagrangian for Eq. (1) is

(7)

Therefore, the adjoint equation E* to Eq. (1) is

(8)

Upon setting v = u it becomes

Hence, Eq. (1) is self-adjoint if and only if it has the form

a = 2b. (9)

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

a = 2b, v = −λu + ε (10)

In this section, we will perfom Lie group method for Eq. (1) on

where ε ≤ 1 the group parameter and and 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

with coefficent

(11)

where D_{k} is the total derivative with respect to independent variables. The invariance condition (6) for Eq. (1) is given by,

(12)

whenever E = 0. The condition (12) is equvalent to

(13)

whenever E = 0. Substituting (11) into (13), yields the determining equations. There are three cases to consider:

**a and b ≠ 0 are arbitrary constants**

In this case, complete set of determining equation is:

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

With substituting (14) – (17) into (18) – (23) we have

(26)

With substituting (26) into (24) – (25) we have

(27)

where c_{1}, c_{2} and K are arbitrary constants. With substituting (27) into determining system, we have

where c_{i}, i = 1,2,3 are arbitrary constants.

**Theorem 3.1.1.** *Infinitesimal generators of every one parameter Lie group of point symmetries in this case are:*

We want to construct the conservation law associated with the symmetry

We have

(28)

The right-hand side of (4) is written

(29)

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

(30)

and

(31)

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:

(32)

Where

**a is an arbitrary nonzero constant and b = 0.**

In this case Eq.(1) is not self adjoint because a ≠ 2b. Complete set of determining equation is:

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

Now, by considering Eq. (33) – (42) it is not to hard to find that the components ξ, τ and φ of infinitesimal generators become

(45)

To find complete solution of the above system, we consider Eq. (43) and

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.

Case 3.2.i). With substituting f = constant in determining system (33)-(44), we have φ = c_{1}, τ = c_{2}, ξ = c_{3}, where c_{i}, i = 1,2,3 are arbitrary constants.

**Theorem 3.2.1.** *Infinitesimal generators of every one parameter Lie group of point symmetries in this case are:*

**Case 3.2.ii). **With integrating from* f _{u} = αf + β* with respect to u, we obtain

(46)

where C is an integrating constant. With substituting (46) into Eq. (43)-(44) and Eq. (45), we have

(47)

**Theorem 3.2.2. ***Infinitesimal generator of every one parameter Lie group of point symmetries in this case is:*

(48)

**Case 3.2.iii).** The Eq. (43) leads to φ = 0, τ = c_{1}, ξ = c_{2}.

**Theorem 3.2.3.** *Infinitesimal generators of every one parameter Lie group of point symmetries in this case are:*

**b = 0, a = 0. **

Complete set of determining equation is

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

To find a complete solution of the above system we consider Eq. (58) and with assumption *f / f _{u} ≠ 0* we rewrite:

(59)

Two general cases are possible:

where c is constant.

**Case 3.3.i).**

With integrating from *f / f _{u} ≠ c* with respect to u, we have

(60)

where L is an integrating constant. Then the Eq. (58) reduce to

(61)

With substituting Eq. (61) into determining equation, we have

(62)

where c_{i}, i = 1,2,3 are arbitrary constants.

**Theorem 3.3.1. ***Infinitesimal generators of every one parameter Lie group of point symmetries in this case are:*

We want to construct the conservation law associated with the symmetry

We have

(63)

The right-hand side of (4) is written

(64)

(65)

Substituting in (64) and (65), the expression (7) for L and (63) for W, we obtain

(66)

(67)

We can eliminate *u _{t}* by using Eq. (1) and obtain

(68)

(69)

Now, we substitute in (68) and (??) the expression v = u, therefore arrive at the conserved vector with the following components:

(70)

(71)

where

**Case 3.3.ii).** By considering Eq. (49) − (54), we find that the components ξ , τ and φ are ξ = ξ(x), τ = τ (t) and . By considering Eq. (55) and (56) we have

By considering Eq. (57) we have

where c_{i}, i = 1..6 are arbitrary constants.

From the following identity:

we find that c_{1} = c_{2} = 0 and . Hence we have two particular cases:

where K is an arbitrary nonzero constant. For the first case, we have

and for the second case, we have

**Theorem 3.2.** *Infinitesimal generators of every one parameter Lie group of point symmetries in this case, when are*

and when are

where K is an arbitrary nonzero constant.

To construct the conservation law associated with the symmetry , we find that . Therefore, we have the conserved vector with the following components

where

The authors wish to express their sincere gratitude to Prof. N.H. Ibragimov for his useful advise and suggestions and helpful comments.

- IbragimovNH (2007) A new conservation theorem. J Math Anal Appl 333: 311-328.
- Ibragimov NH, Khamitova RS, Valenti A (2011) Self-adjointness of generalized Camassa-Holm equation. J Applied Mathematics and Computation 218: 2579-2583.
- Clarkson PA, Mansfield EL, Priestley TJ (1997) Symmetries of a Class of Nonlinear Third Order Partial Differential Equations. Math Comput Modelling 25: 195-212.
- Nadjafikhah M, Shirvani-Sh V (2011) Symmetry classification and conservation laws for higher order Camassa-Holm equation.
- IbragimovNH(2007) Quasi-self-adjoint differential equations. Arch ALGA 4: 55-60.
- OlverPJ (1986) Applications of Lie Group for Differential Equations. Springer-Verlag, New York.
- OlverPJ (1995) Equivalence, invariant and symmetry. Cambridge University Press, Cambridge.

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