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Self-adjointness, Group Classification and Conservation Laws of an Extended Camassa-Holm Equation | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Self-adjointness, Group Classification and Conservation Laws of an Extended Camassa-Holm Equation

Nadjafikhah M1* and Pourrostami N2

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

2Department 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
Tel:
+98 2173225426
Fax:
+982173228426
E-mail:
[email protected]

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.

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Abstract

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.

Keywords

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

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

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

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

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

Preliminaries

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 image Here

image

is the variational derivative and Di 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 image such thatimage with an undetermined coefficient λ. Eq.(1) is said to have a nonlocal conservation law if there exits a vector C = (C1, C2) satisfying the equation

image (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 image is obtained by the following formula :

image (4)

where i, j, k = 1,2 and imageimage

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

image (5)

involving image and the derivatives of u with respect to x up to n, where image 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):

image where imageimage 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 image 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]).

 

image (6)

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

Adjoint Equation and Classical Symmetry Method

Formal Lagrangian for Eq. (1) is

image (7)

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

image (8)

Upon setting v = u it becomes

image

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

a = 2b. (9)

Consider again Eq. (1), and substitute

image

in the adjoint equation (8), then

image

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

image where ε ≤ 1 the group parameter and image and image are the infinitesimals of the transformations for the independent and dependent variables respectively. The associated vector fields is of the formimage and the third porolongation of v is the vector field

 

image

with coefficent

image (11)

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

image (12)

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

image (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:

image (14)
image (15)
image (16)
image (17)
image (18)
image (19)
image (20)
image (21)
image (22)
image (23)
image (24)
image (25)

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

image (26)

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

image (27)

where c1, c2 and K are arbitrary constants. With substituting (27) into determining system, we have

image

where ci, 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:

image

We want to construct the conservation law associated with the symmetry

image

We have

image (28)

The right-hand side of (4) is written

image (29)

We eliminate the term ξiL 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

image (30)

and

image (31)

We can eliminate ut 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:

image (32)
image

Where image

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:

image (33)
image (34)
image (35)
image (36)
image (37)
image (38)
image (39)
image (40)
image (41)
image (42)
image (43)
image (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

image (45)

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

3.2.i) l = 1, then f = constant;

3.2.ii) l = 2, then fu = αf + β ;

3.3.iii) l = 3, then αfu + βf + γ ≠ 0, α ≠ 0.

Case 3.2.i). With substituting f = constant in determining system (33)-(44), we have φ = c1, τ = c2, ξ = c3, where ci, 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:

image

Case 3.2.ii). With integrating from fu = αf + β with respect to u, we obtain

image (46)

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

image (47)

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

image (48)

Case 3.2.iii). The Eq. (43) leads to φ = 0, τ = c1, ξ = c2.

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

image

b = 0, a = 0.

Complete set of determining equation is

image (49)
image (50)
image (51)
image (52)
image (53)
image (54)
image (55)
image (56)
image (57)
image (58)

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

image (59)

Two general cases are possible:

image

where c is constant.

Case 3.3.i).

With integrating from f / fu ≠ c with respect to u, we have

image(60)

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

image(61)

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

image(62)

where ci, 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:

image

We want to construct the conservation law associated with the symmetry

image

We have

image (63)

The right-hand side of (4) is written

image (64)
image (65)

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

image (66)
image (67)

We can eliminate ut by using Eq. (1) and obtain

image (68)
image (69)

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

image (70)
image (71)

where image

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 image . By considering Eq. (55) and (56) we have

image
image

By considering Eq. (57) we have

image

where ci, i = 1..6 are arbitrary constants.

From the following identity:

image

we find that c1 = c2 = 0 and image. Hence we have two particular cases:

image

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

image

and for the second case, we have

image

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

image

and when image are

image

where K is an arbitrary nonzero constant.

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

image

where image

Acknowledgements

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

References

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