Exact solutions of the classical Boussinesq system

Abstract In this paper, we study exact solutions of the classical Boussinesq (CB) system, which describes propagations of shallow water waves. By using the bilinear form, with exponential expansions, we obtain solitary wave solutions of the CB system. Based on asymptotic analysis method, we study the elastic and elastic–inelastic–coupled interactions of the obtained solitary wave solutions. With extended three–wave method, we obtain the periodic solitary solution of the CB system. And with polynomial expansions, we get the rational solutions of the CB system. These interesting exact solutions may be useful in the study of some phenomena appeared in shallow water waves.

In this paper, based on the bilinear form, we consider exact solutions including solitary wave solution, periodic solitary solution and rational solution of the classical Boussinesq (CB) system (see Wu & Zhang, 1996) (1) where, u is the elevation of the water wave and v is the surface velocity of water along x-direction. This system was introduced in Wu and Zhang (1996), which is derived from Euler equation. This system can be used to study the run-up of ocean waves such as tsunami waves on dykes and dams. A good understanding of exact solutions is helpful in harbour and coastal design. Therefore, finding more type of solutions is very important in fluid dynamics. In Li, Ma, and Zhang (2000); Zhang (2001, 2003); Zhang, Chang, and Li (2009); Zhang and Li (2003); Zhang, Zhao, and Chen (2015), by using Darboux transformation, the authors obtained the bidirectional solitons on water and the elastic-fusion-coupled interaction of the CB system (1). In Roshid and Rahman (2014); Zayed and Shorog (2010); Zhang et al. (2002), by using of Tanh function, extended ðG 0 =GÞ-expansion and expðÀUðgÞÞexpansion, the authors obtained the traveling wave solutions including hyperbolic solution, trigonometric solution, periodic solution of the CB system (1).
Through the following proper transformation (2) the CB system (1) can be transformed into the bilinear form In Section 2, based on the bilinear form in (3) and assuming f and g have exponential expansions, we can obtain solitary wave solutions of the CB system (1), including the bidirectional soliton and the elastic-inelastic-coupled solitary wave solution. By using asymptotic analysis method (Chakravarty & Kodama, 2008;Lambert, Musette, & Kesteloot, 1987;Wang, Tian, Li, Wang, & Jiang, 2014;Zhang & Chen, 2016), we analyze the interactions of the obtained solutions. In Section 3, by using extended three-wave method, we obtain single soliton solution, periodic solitary wave solution, and solitary wave solution with fission of the CB system (1). In Section 4, taking f and g as polynomial expansions (L€ u et al., 2016a, 2016bShi et al., 2015;Zhang & Ma, 2015), we get rational solutions of the CB system (1). In Section 5, we give our conclusions.

Solitary wave solutions of the CB system
In this section, based on parameter expansions and taking f and g as exponential forms, solitary wave solutions of the CB system (1) are obtained according to the bilinear form (3). Then by using asymptotic analysis method, we study propagations and interactions of the solitary wave solutions of the CB system (1).
In order to get the bilinear derivative Equation (3), we assume that f and g have the following asymmetric form expansions of the bookkeeping parameter e g ¼ eg 1 ðx; tÞ þ e 2 g 2 ðx; tÞ þ e 3 g 3 ðx; tÞ þ Á Á Á : Substituting (4) and (5) into the bilinear form (3), k i and a i ði ¼ 1; 2; Á Á Á ; NÞ are arbitrary constants. Substituting (6) into the transformation (2), we can obtain the multi-solitary wave solutions of the CB system (1). In the following discussions, we give detailed analysis for the cases of N ¼ 1, 2, 3. For N ¼ 1, based on (6), we obtain Substituting f and g into (2), we get the single soliton solution of (1) where, u is a bell-soliton solution and v is a kink-soliton solution with velocity k 1 =2 (see Figure 1).
of the solution u through the asymptotic analysis method. The analysis for v is similar and is omitted for brevity. Without loss of generality, we let 0 < k 1 < k 2 and we have the following asymptotic expressions.

Periodic solitary wave solutions of the CB System
In this section, based on the bilinear form (3), by using the extended three-wave method, we find the single soliton solution, periodic solitary solution and solitary wave solution of the CB system (1).
We formulate solutions of the bilinear form (3) as f ¼ a 1 e g 1 þ a 2 cosg 2 þ a 3 coshg 3 þ e Àg 1 ; where, g i ¼ k i x þ w i t, i ¼ 1, 2, 3. Substituting (23) into the bilinear form (3) and equating the coefficients of e Àg 1 cosg 2 ; e Àg 1 coshg 3 , Á Á Á to zeros, we obtain a system of algebraic equations with the unknowns of a i , k i , w i (i ¼ 1, 2, 3) and b j (j ¼ 1; 2; 3; 4) which can be solved by using Mathematica. And we have the following different sets of solutions.

Rational solutions of the CB system
In this section, based on polynomial solutions of the bilinear form (3), we consider rational solutions of the CB system (1).

Conclusions
In this paper, based on the bilinear form, via the exponential expansion, the expanded three-wave method and polynomial expansion, explicit exact solutions of the classical Boussinesq system are derived, which include solitary wave solution, periodic solitary solution and rational solution. By using of the asymptotic analysis method, the interactions of the obtained solitary wave solutions are discussed in detail. In addition, plots of the obtained solutions are revealed with the help of Mathematica. These Figure 5. Plots of the solitary wave solution (29) of the CB system (1) with k 1 ¼ 1; k 3 ¼ 4; b 3 ¼ 1; b 4 ¼ 5.  (27) of the CB system (1) with k 1 ¼ À1; k 2 ¼ 3, b 2 ¼ À2; b 4 ¼ 9.
interesting exact solutions may be useful in the study of some phenomena appeared in shallow water waves.