Exact Solutions of Generalized Modified Boussinesq , Kuramoto-Sivashinsky , and Camassa-Holm Equations via Double Reduction Theory

We find exact solutions of the Generalized Modified Boussinesq (GMB) equation, the Kuromoto-Sivashinsky (KS) equation the and, Camassa-Holm (CH) equation by utilizing the double reduction theory related to conserved vectors. The fourth order GMB equation involves the arbitrary function and mixed derivative terms in highest derivative. The partial Noether’s approach yields seven conserved vectors for GMB equation and one conserved for vector KS equation. Due to presence of mixed derivative term the conserved vectors for GMB equation derived by the Noether like theorem do not satisfy the divergence relationship. The extra terms that constitute the trivial part of conserved vectors are adjusted and the resulting conserved vectors satisfy the divergence property. The double reduction theory yields two independent solutions and one reduction for GMB equation and one solution for KS equation. For CH equation two independent solutions are obtained elsewhere by double reduction theory with the help of conserved Vectors.


Introduction
Nonlinear differential equations have many significant implications for the mathematical models and have been of great interest in the last few decades.In the literature various techniques are used to construct exact and numerical solutions of differential equations [1][2][3][4][5][6].Different approximations and numerical methods can be implemented to reduce nonlinear partial differential equations (PDEs) to ordinary differential equations (ODEs) but the problem occurs in the convergence of the solutions.Symmetry analysis and conservation laws play a vital role in the analysis of differential equations and have many important applications in numerical methods, linearization, and integrability.For variational type differential equations standard Lagrangian always exists and conservation laws can be computed from the well-known Noether's formula [7].The standard Lagrangian does not exist for nonvariational differential equations.Kara and Mahomed [8] introduced the notion of partial Lagrangians for such differential equations.A Noether like theorem was invoked for the derivation of conservation laws associated with a partial Lagrangian.This technique is referred as to Noether's like approach or partial Lagrangian approach.There are some approaches for the construction of conservation laws which do not use the knowledge of standard or pariah Lagrangians.The different methods to compute the conservation laws and the comparison of approaches were studied in [9] (also see references therein).
The relationship between Noether symmetries and conservation laws is well known and is useful to reduce the numbers of variables and order of differential equations.In [10], a conserved vector is associated with the Lie-Bäcklund symmetries.Sjöberg and Mahomed [11,12] have generalized association of a conserved vector to nonlocal symmetries.The association of Lie-Backlund symmetries or nonlocal symmetries with a conserved vector led to the development of the double reduction theory for the nonvariational type PDEs or system of PDEs with two independent variables [13,14].Bokhari et al. [15] generalized the theory of double reduction to find the invariant solutions for a nonlinear system of PDEs with several independent variables.This theory is helpful in finding invariant solution of PDEs having a nontrivial conserved vector and at least one symmetry associated with it.Recently, Narain and Kara [16] redefine the variational and nonvariational approaches for a class of PDE involving mixed derivative terms.Due to the presence of mixed derivative term a conserved vector computed by Noether's theorem does not satisfy the divergence relationship.A number of extra terms contributing to the trivial part of conserved vector arise and need to be adjusted to satisfy the divergence relationship.
The objective of this paper is to find the exact solutions of GMB, KS, and CH equations using the double reduction theory.The conservation laws of GMB and KS equations are constructed via partial lagrangian approach.For GMB equation the derived conserved vector failed to satisfy the divergence property.The extra terms which constitute the trivial part of the conserved vectors are adjusted to satisfy the divergence relationship.After construction of conservation laws the theory of double reduction is applied to compute the solutions of GMB and KS equations.The conserved vectors of Camassa-Holm equation are derived in [17].Two exact solutions are computed for the Camassa-Holm equation by utilization of double reduction theory.
This paper is arranged in the following manner.In Section 2 , basic definitions, fundamental operators, and theorem for double reduction theory are invoked.In Section 3, the conservation laws and exact solutions for GMB equation are presented.The conservation laws and solution for KS equation are derived in Section 4. Section 5 deals with the exact solutions of CH equation using double reduction theory.Concluding remarks are summarized in Section 6.
Definition 4. A Lie-Bäcklund operator  defined in ( 6) is said to be a Noether symmetry generator corresponding to a Lagrangian  = (, ,  (1) , . . .,  () ) of ( 1) if it satisfies where   is the total derivative operator defined in (2) and   are the guage terms.The generator  in ( 6) is said to be a partial Noether operator corresponding to a partial Lagrangian  = (, ,  (1) , . . .,  () ) of (1) if Definition 5.If  in ( 6) is a Noether symmetry with respect to a Lagrangian or partial Noether operator with respect to a partial Lagrangian, then the conserved vector can be constructed from Definition 6. Assume that  is a symmetry of the system (1) and  is a conserved vector of (1).If  and  satisfy Then  is associated with .
The following theorem illustrates the construction of new conservation laws from symmetries and known conservation laws [18,19].
We now describe some results which are used in our work (see [13,14]).
Consider a scalar partial differential equation  = 0 with  = 2, ( 1 ,  2 ) = (, ) such that it has conserved vector (  ,   ) and admits symmetry generator  associated with conserved vectors.In terms of canonical variables ,  with the symmetry  = / the conservation laws can be expressed as     +     = 0.The vectors   and   in terms of (, ) are Theorem 8.A PDE of order  with two independent variables, which admits a symmetry  that is associated with a conserved vector , is reduced to an ODE of order  − 1; namely,   = , where   is defined in (15) for solutions invariant under .

Conservation Laws and Exact Solutions of GMB Equation
The generalized modified Boussinesq (GMB) equation is where  is a constant and () is an arbitrary function.GMB equation describes nonlinear model of wave propagation of elastic rods and also arises in nonlinear lattice waves, iron sound waves, and vibrations in a nonlinear string and is thus important to study.
Equation ( 17) admits a partial Lagrangian and the corresponding partial Euler-Lagrange equation is Substituting these values in (10) and comparing the coefficients of like monomials of  we obtain In order to solve the system (20), we discuss the following cases.
Substituting the above values of , , ,  1 , and  2 in (11) we obtain where  1 , . . .,  4 are arbitrary constants.The choice of constants one by one equal to one and the rest to zero yields the following conserved vectors: The divergence of (22) becomes The conserved vectors in (22) fail to satisfy the divergence properties.Narain and Kara [16] prove that   can be adjusted to T such that   T = 0. Following the same line we find that the modified conserved vectors T  ( = 1, 2,  = 1, 2, 3, 4) are Case 2.   ̸ = 0.In this case after some simple but lengthy manipulations result in  = 0,  = 0,  =  1  +  2 +  3 ,  1 = − 3 and  2 = () 1 .Substitutig , , ,  1 , and  2 in (11), we arrive at The choice of constants gives rise to The conserved vectors in (26) do not satisfy the relation     = 0; that is, The new conserved vectors T after adjustment result in Now, we apply the double reduction theory to associate the Lie point symmetries with the conservation laws.Equation ( 17) admits the following Lie point symmetries when   = 0; that is, () =  + .Consider where (, ) satisfies (17).
The symmetries  1 and  2 in (29) are associated with the conservation laws of GMB equation given below when   = 0: The combination of these symmetries  =  1 +  2 yields the generator,  in canonical form  = / if or Using relation (15), we determine   1 and   2 corresponding to  1 and  2 given in (30): Since The solution of ( 35) is where and  1 ,  2 , and  3 are arbitrary constants.Hence is a solution of ( 17) invariant under .
Similarly from (34), one can easily find that is also a solution of (17).Equation ( 17) admits the trivial symmetry generators  1 = / and  2 = / in Case 2 when () is an arbitrary function.In this case the symmetry generators  1 = / and  2 = / are associated with the conserved vectors of GMB equation.The similarity variables are defined in (32) and with the help of conserved vectors (40), ( 17) reduces to Equation ( 41) cannot be reduced further due to an arbitrary function ().

Conservation Laws and Exact Solution of the Kuramoto-Sivashinsky Equation
The Kuramoto-Sivashinsky (KS) equation plays a dominant role in stability of flame fronts, reaction diffusion, and other physical phenomena.The KS equation is where , , and  are constants.Equation (42) does not admit a standard Lagrangian but the partial Lagrangian is and the corresponding partial Euler-Lagrange equation is Substituting these values in (10) and after some lengthy manipulation results in  = 0,  = 0,  = ,  1 = , and  2 = .Setting  = 1, we obtain  = 0,  = 0,  = 1,  1 = , and  2 = .Equation (11) with the substitution of , , ,  1 , and  2 gives rise to which satisfies We reduce (42) using the double reduction theory, that is, association of symmetries with the conserved vectors.Equation (42) possesses three Lie point symmetry generators: Using (13) one can easily verify that  1 ,  2 in (47) are associated with the conserved vector of KS equation.We set  =  1 +  2 ; then, the canonical coordinates of  are  = ,  =  − , and  = ().Using (15) and with the use of the above canonical coordinates, we find that Replacing   =  (arbitrary constant), (49) becomes Equation ( 50) admits the symmetry generator  = / and the similarity variables are V = ,  =   .After using the similarity variables, (50) reduces to The solution of (51) gives rise to Equation ( 52) in terms of  =   and V =  yields which in turn results in which is a solution of (42).

Exact Solutions of the Camassa-Holm Equation
The Camassa Holm (CH) equation is described by Camassa and Holm as a bi-Hamiltonian model for waves in shallow water.It is prominent for turbulent flows and waves in a hyperelastic rod.The CH equation is where  and  are constants.Conservation laws of CH equation are derived in [17]: Equation ( 55) admits the following symmetries: From ( 13) it can be easily shown that the conserved flow ( 1 ,  2 ) for CH equation are associated with only  1 ,  2 .We define the combination of these symmetries  =  1 +  2 .
The generator  has the canonical form  = / if Equation ( 15) with the use of similarity variables defined above reduces (56) to Setting   1 =  in (60), we obtain The solution of (62) yields which is a solution of (55).
Similarly from (61), applying the same procedure, we have which constitutes the solution of (55).

Conclusions
Exact solutions of the GMB equation, the KS equation, and the CH equation were constructed by utilizing the conservation laws.Firstly GMB equation was considered and conservation laws were computed by partial Noether's approach.Two cases arise; namely, Case 1:   = 0 and Case 2:   ̸ = 0.In Case 1, when () was a linear function four conserved vectors were obtained, whereas in Case 2 for arbitrary () three conserved vectors were reported.The derived conserved vectors failed to satisfy divergence condition.The extra terms arising in conserved vectors were absorbed and the new forms of conserved vectors satisfying the divergence property were found.When () is linear only two conserved vectors (30) satisfy the symmetry conservation laws relationship.The double reduction theory was applied to these two conserved vectors and two independent solutions were constructed.The symmetry was associated with only one conserved vector (40) when () is arbitrary.For this case GMB equation was reduced to a second order ODE.The partial Noether approach for KS equation yielded one conserved vector which satisfies the symmetry conservation laws relation.The conserved vector reduced the KS equation to a third order ODE (50) which further reduced to a second order ODE (51) which in turn results in the exact solution (54) of KS equation.A similar procedure is carried out to obtain two exact solutions of CH equation.These solutions are new and not obtained in the literature.The derived solutions cannot be interpreted physically due to deficiency of experimental sources; however these are important for numerical simulations.