Abstract
It is proved that the modified Boussinesq equation is consistent Riccati expansion (CRE) solvable; two types of special soliton-cnoidal 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.
1. 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 [18–22].
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: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.
2. 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 formwhere , , and are undetermined functions of space-time and and satisfies the Riccati equationwhich includes 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.
2.2. 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 aswhere , , and are undetermined by (7) and (8) with , , , and . Thus, we haveand the function satisfies
In brief, we arrive at the following nonauto-BT theorem for (1).
Theorem 1. If is a solution of (13), then , withis a solution of modified Boussinesq equation (1).
Theorem 1 shows that the single soliton solution of (1) is only the straight line solution 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 formwhere 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 satisfieswithand , 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.
Case 1. A simple solution of (17) is given aswhich leads to the soliton-cnoidal wave interaction solution of (1) as follows:where are arbitrary constant, , , , andIn solution (21), is the third type of incomplete elliptic integral.
The dynamic behavior of soliton-cnoidal wave interaction solution (21) with the parameters , , , and 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].
(a)
(b)
Case 2. Another special solution of (17) reads as follows:which yields the soliton-cnoidal wave interaction solution of (1):where are arbitrary constant, , , , and
In Figure 2, we plot soliton-cnoidal wave interaction solution (24) with the parameters selected as , , and . It presents that a soliton propagates on a cnoidal wave background instead of propagating on the plane continuous wave background.
(a)
(b)
3. Nonlocal Symmetry and Its Localization
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.
Theorem 2. If is a solution of (13), then , withis 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 transformationwith the infinitesimal parameter .
Proposition 3. Modified Boussinesq equation (1) has a nonlocal symmetry given bywhere satisfies (13).
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 findwhich 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 aswhere is the singular manifold and and are functions of to be determined later.
Substituting (31) into (6) and vanishing coefficients of the each powers of , we obtainConsequently,is a solution of (1) with satisfying Schwartzian form of (1):where the notations and are defined asSchwarzian equation (34) is invariant under the Möbious transformationwhich means (34) possesses the following Lie point symmetries in the form ofwith , , and 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), thenis 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, , is a symmetry of (1) with the solution . 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.
3.2. Localization of Residual Symmetry
To find out the symmetry group transformation of nonlocal residual symmetry, we should solve the following initial value problem:where is an arbitrary group parameter.
Due to the intrusion of the function 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 symmetryfor the prolonged systemProlonged system (41) is closed after covering dependent variables , , and with the vector formCorrespondingly, initial value problem (39) is changed asBy 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 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 setthe first equation of (44) is just the truncated Painlevé expansion (31) with (32).
4. 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 , 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.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to Professor Y. Chen for his helpful suggestions and fruitful discussion. This work is supported by National Natural Science Foundation of China under Grant nos. 11271211, 11435005, and 11505090, K. C. Wong Magna Fund in Ningbo University, and Research Award Foundation for Outstanding Young Scientists of Shandong Province (no. BS2015SF009).