CRE Solvability, Exact Soliton-Cnoidal Wave Interaction Solutions, and Nonlocal Symmetry for the Modified Boussinesq Equation

It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special solitoncnoidal wave interaction solution of the equation are explicitly given, which is difficult to be found by other traditional methods. Moreover, the nonlocal symmetry related to the consistent tanh expansion (CTE) and the residual symmetry from the truncated Painlevé expansion, as well as the relationship between them, are obtained.The residual symmetry is localized after embedding the original system in an enlarged one.The symmetry group transformation of the enlarged system is derived by applying the Lie point symmetry approach.


Introduction
The exact solutions for nonlinear evolution equations (NLEEs) arising from many science fields are important because of their wide applications in explaining physical phenomena.Many powerful methods for obtaining the exact solutions of NLEEs have been presented, such as the inverse scattering transformation [1], the Darboux transformations (DT) [2], the Bäcklund transformation (BT) [3], the Lie group method [4,5], Hirota's bilinear method [6], the Painlevé analysis [7], separated variable method [8], homogeneous balance method [9], and tanh function expansion method [10].For these methods, the interaction solutions among different nonlinear excitations such as solitons on a cnoidal wave background are difficult to obtain.The soliton solutions on the background of the periodic solutions which are given by the elliptic theta function can be found with the help of Cauchy-Baker kernel [11,12].
Recently, Lou found that the symmetry from the truncated Painlevé expansion is just the residue with respect to the singular manifold and proposed residual symmetry [13,14].Furthermore, by developing the truncated Painlevé expansion, Lou [15,16] established the consistent Riccati expansion (CRE) method, which is a simple but effective method to construct interaction solutions among different nonlinear excitations.The method is valid for some nonlinear models like the bosonized supersymmetric Korteweg-de Vries (KdV) model [14], the nonlinear Schrödinger system [16], the Broer-Kaup System [17], and so on [18][19][20][21][22].
In this paper, we focus on investigating the CRE solvability and nonlocal symmetry of the modified Boussinesq equation: which was proposed by Hirota and Satsuma [23] from a BT of the Boussinesq equation: 2

Advances in Mathematical Physics
Equation ( 1) is called the modified Boussinesq equation since it is linked with the known Boussinesq equation ( 2) by the Miura transformation: The generalized Hamiltonian form and the corresponding finite-dimensional integrable systems of (1) are obtained with the help of the nonlinearization approach of Lax pairs [24].The Painlevé property [25] and similarity solutions [26,27] have been studied.To our knowledge, the analytic interaction solutions among different nonlinear excitations for (1) have not been obtained up to now.Here we will study the interaction solution between the soliton and the cnoidal periodic wave of (1).The paper is arranged as follows.Section 2 is devoted to the CRE method for the modified Boussinesq equation.As a result, two types of special explicit interaction solution between the soliton and the cnoidal periodic wave of this equation are given.In Section 3, for the modified Boussinesq equation, we derive the nonlocal symmetry related to the CTE and the nonlocal residual symmetry from the truncated Painlevé expansion.The relationship between them is presented.Then the corresponding symmetry group transformation is found in the process of the localization of residual symmetry.The last section is a short summary and discussion.

CRE Solvability and Soliton-Cnoidal
Wave Interaction Solutions 2.1.CRE Solvability.Based on the CRE method in [15], the possible truncated expansion expression for (1) has the form where  0 ,  1 , and  are undetermined functions of spacetime  and  and () satisfies the Riccati equation which includes tanh() as a special case.Differentiating with respect to , (1) becomes Substituting (4) with ( 5) into (6) and vanishing all the coefficients of the powers of (), we obtain ten overdetermined equations for only three undetermined functions.Fortunately, these overdetermined equations are consistent and possess the following solution: and the function  satisfies a generalization of the Schwartzian form of (1): where Due to consistency of the overdetermined system, we call the modified Boussinesq equation CRE solvable.

CTE Solvability and Soliton-Cnoidal Wave Interaction
Solutions.We consider the function () in Riccati equation ( 5) as the following special solution: which means that a CRE solvable system must be consistent tanh expansion (CTE) solvable, and vice versa.Meanwhile, truncated expansion expression ( 4) is changed as where  0 ,  1 , and  are undetermined by ( 7) and ( 8) with  0 = 1,  1 = 0,  2 = −1, and  = 4. Thus, we have and the function  satisfies In brief, we arrive at the following nonauto-BT theorem for (1).

Theorem 1.
If  is a solution of ( 13), then , with is a solution of modified Boussinesq equation (1).
Theorem 1 shows that the single soliton solution of ( 1) is only the straight line solution  =  0  +  0  of (13); the interaction solutions between solitons and other nonlinear excitations of (1) can be constructed by solving (13).To find the interaction solutions between one soliton and other nonlinear waves of (1), we consider  in the form where  is a function with respect to  and .In this study, we only discuss the solutions with the form Substituting ( 16) into (13), we can find that  1 () satisfies with and  1 ,  2 are arbitrary constants.Then (1) has the explicit solution expressed as It is clear that (17) has abundant explicit solutions in terms of Jacobi elliptic functions.Hence, solution (19) exhibits the interactions between one soliton and cnoidal periodic waves.In what follows, only two nontrivial cases are considered in detail to obtain this kind of solution.
The dynamic behavior of soliton-cnoidal wave interaction solution (21) with the parameters  0 = 1,  1 = 1.2,  = 0.9, and  = 0.5 is illustrated in Figure 1.It can be seen from Figure 1 that a soliton moves on a cnoidal wave background instead of moving on the plane continuous wave background.This kind of solution can be applicable to describe some interesting physical phenomena, such as the Fermionic quantum plasma [28].

Nonlocal Symmetry and Its Localization
is a solution of modified Boussinesq equation (1).
Proof.By direct substitution, a symmetry   of ( 6) is defined as a solution of its linearized equation, which means ( 6) is form invariant under the infinitesimal transformation with the infinitesimal parameter .
Proof.By direct calculation, we substitute (28) into the linearized equation of ( 6) with the help of the nonauto-BT (26) in Theorem 2 and the  equation ( 13).Now, applying the following transformation: to the nonlocal symmetry (28), we find which is the residual symmetry of (1).
Here we can obtain residual symmetry (30) from the truncated Painlevé expansion.Balancing the nonlinear and dispersive terms in (1), the truncated Painlevé expansion can be written as where  is the singular manifold and  0 and  1 are functions of (, ) to be determined later.Substituting (31) into ( 6) and vanishing coefficients of the each powers of 1/, we obtain Consequently, is a solution of (1) with  satisfying Schwartzian form of (1): where the notations  and  are defined as Schwarzian equation ( 34) is invariant under the Möbious transformation which means (34) possesses the following Lie point symmetries in the form of with  1 ,  1 , and  1 being arbitrary constants.From the above standard truncated Painlevé expansion, we have the BT theorem as follow.
Theorem 4. If  is a solution of the Schwartzian equation ( 34), then is a nonauto-BT between  and the solution  of modified Boussinesq equation (1).
The residual of truncated Painlevé expansion (31) with the singular manifold , that is,  0 , is a symmetry of (1) with the solution  1 .Thus, (30) is the residual symmetry of (1).It should be emphasized that the above residual symmetry is just related to the Möbious transformation symmetry (37) by the linearized equation of nonauto-BT (38) in Theorem 4.

Localization of Residual
where  is an arbitrary group parameter.Due to the intrusion of the function φ() and its differentiations, it is difficult to solve the above initial value problem according to the Lie first theorem.Thus, one may extend the original system such that nonlocal residual symmetry becomes the local Lie point symmetry of a closed prolonged system.
The nonlocal residual symmetry of (1) can be localized to the Lie point symmetry for the prolonged system Prolonged system (41) is closed after covering dependent variables , , and  with the vector form Correspondingly, initial value problem (39) is changed as By solving the above initial value problem, we arrive at the symmetry group transformation theorem as follow.
Theorem 5.If {, , } is a solution of prolonged system (41), then so is {û, ĝ, φ} with Finite transformation (44) provides a way to generate a new solution from old one.It is necessary to point out that the nonlocal residual symmetry is nothing but the infinitesimal form of the finite transformation.Actually, if we set the first equation of ( 44) is just the truncated Painlevé expansion (31) with (32).

Summary and Discussion
In conclusion, the modified Boussinesq equation is proved to be CRE integrable.The CTE method which is a special simplified form leads to a nonauto-BT, which strengthens a single soliton to a straight line solution.Two types of special interaction solution between the soliton and the cnoidal periodic wave are derived by means of the nonauto-BT.On the other hand, for the modified Boussinesq equation, the nonlocal symmetry related to the CTE is derived.Under the transformation  = 1/(1 − tanh()), this nonlocal symmetry is changed as the residual symmetry which can be obtained from the truncated Painlevé expansion.To solve the initial value problem related to the residual symmetry, we extend the original system such that residual symmetry becomes the local Lie point symmetry of a closed prolonged system.The symmetry group transformation of the prolonged system is derived by using the Lie first theorem.
More about further integrable properties from CRE will be investigated in detail in our future research work.

Figure 1 :
Figure 1: Interaction between the soliton and the cnoidal periodic wave via solution (21) with the parameters  0 = 1,  1 = 1.2,  = 0.9, and  = 0.5.(a) The profile of the special structure at  = 0. (b) Overhead view of the wave.

3. 1 .
Nonlocal Symmetry.To derive the nonlocal symmetry related to the CTE, we give a nonauto-BT theorem for modified Boussinesq equation (1) as follows.
Symmetry.To find out the symmetry group transformation of nonlocal residual symmetry, we should solve the following initial value problem: