1 Introduction

We consider the following family of Boussinesq type systems of water wave theory, model by Zhang et al. ^[1] and Chen et al. ^[2]

where v is the elevation of a water wave and u is the surface velocity of water along x-direction and c1=-(1/2)(ϕ2-(1/3)), c2=(1/2)(1-ϕ2) and ϕ is a depth of water (ϕ=0 is at the bottom, ϕ=1 is on the surface), which have the relation c1-c2=1/3. For the case c1=0,  c2=-1/3, (1) is the classical Boussinesq (cB) system is not linearly well posed in the Hadamard sense ^[3], it is important because it has an integrable Hamiltonian structure ^[4] and exact solitary-wave solutions ^[5-7].

This (1) is also known as Nwogu’s Boussinesq (NB) model is useful for coastal and civil engineering to perform the nonlinear water wave model in a harbour and coastal design. Therefore many scientists studied mathematical properties, such as bifurcation and travelling wave solutions, lie symmetry analysis, single and multiple solitary wave solutions and painleve analysis ^[8-16].

In recent years, many methods have been developed for the exact solutions of nonlinear evolution equations such as the sub-equation method, the modified trial equation method, the simplest equation method, the generalized Kudryashov method, the symmetry analysis method and so on ^[17-22]. The main objective of this study is to investigate new traveling wave solutions for NB model by the Jacobi elliptic function method. The effectiveness and efficiency of this method are shown in literature with the various Jacobi elliptic function forms ^[23-27].

The outline of the present paper is as follows. In Sec. 2, we have a brief description of the Jacobi elliptic function method for solving partial differential equations. In Sec. 3, we apply the Jacobi elliptic function method above mentioned equation. Finally, some conclusions are given the latest section.

2 Jacobi’s elliptic function method

In this section, we would like to describe extended Jacobi elliptic function method. Suppose a nonlinear partial differential equation (NPDE) with independent variables x,t and dependent variable u:

Considerthe following travelling wave transformation

where c is an arbitrary constant to be determined later. By substituting (3) into (2), we have an ordinary differential equation (ODE):

Let us consider the solutions in the form

where F satisfies the (2) and   n   is a positive integer which can be evaluated by balancing the highest order partial derivative term and nonlinear term in (2) or (4). F(ξ) satisfies the following auxiliary equation:

where P, Q, and R   are constants. The last equation hence holds for F(ξ):

With the help of Maple, substituting (6) into (4) along with (7) and collecting the coefficients of the same power Fi(F')j  (j=0,1,  i=0,1,2,...) and setting each of the attained coefficients to be zero we have a set of over determined algebraic equations. And after we solve this by Maple, we find P, Q, R and c. Substituting the attained results into (6), gives the exponential and periodic solutions. It is well-known that (6) has families of Jacobi elliptic function solutions as follows ^[28,29]:

In this Table snξ,  cnξ,  dnξ are respectively Jacobian elliptic sine function, Jacobian elliptic cosine function and the Jacobian elliptic function and the other Jacobian functions can be generated by these three kinds of functions, namely

Also these functions satisfying the following formulas:

And addition derivative properties,

The Jacobian-elliptic functions degenerate into hyperbolic functions when m→1 as follows:

The Jacobian-elliptic functions degenerate into trigonometric functions when m→0 as follows:

3 Generalized Boussinesq System (GBQS)

Suppose that the travelling wave solutions for Eq. (11) are of the forms as follows:

where c is a constant to be determined later and ξ is an arbitrary constant.

By substituting (10) into (11), we have an ordinary differential equation (ODE):

where prime denotes differentiation with respect to ξ. Now, balancing the nonlinear terms u''' and u'u, we get m=2. Balancing the nonlinear terms u''' and uv', we get n=2. Hence, from (5), we might constitute

in which a0,a1,a2,b0,b1 and b2 are undetermined constants. Substituting (13) and (6) into (12) and setting the coefficients of Fi(ξ)F'(ξ)j=0,  i=0,1,2,3,  j=0,1 to zero yields the following set of algebraic equations for a0,a1,a2,b0,b1, b2 and c:

Solving the set of nonlinear algebraic equations by help of Maple program, the following results are attained.

Substituting these results into (13), we have the following solution of Eq. (16):

With the means of table and from the above solution (16), one might induce more general united Jacobian-elliptic function solutions of Eq. (12). Hereby, we attain the following exact solutions.

In the limit case when m→1, we get the solitary wave solutions of Eq. (12). In the limit case when m→0, we acquire the traveling wave solutions of Eq. (12).

Case 1. If we take P:m2,  Q:-(1+m2), then F(ξ)=snξ, thus

In the limit case when m→1, then F(ξ)=tanhξ, and we attain one of the solitary wave solutions of Eq. (12) as

Case 2. When P:-m2,  Q:(2m2-1) are chosen, then F(ξ)=cnξ, therefore

Considering m→1, then F(ξ)=sechξ, and one of the solitary wave solutions of Eq. (12) has been obtained as

Case 3. Choosing P:1,  Q:-(1+m2), it may be denoted from table F(ξ)=nsξ, hence

As m→1, then F(ξ)=cothξ, and one of the solitary wave solutions of Eq. (12) can be stated as

Case 4. Supposing P:1,  Q:-(1+m2) from table this choices correspond to F(ξ)=nsξ, hence

For m→0 from (9), F(ξ)=cscξ, and we acquire one of the periodic solutions of Eq. (12) as

Case 5. Considering P:1,  Q:-(1+m2) from (9) F(ξ)=dcξ, hence

If m→0, then F(ξ)=secξ, and one of the periodic solutions of Eq. (12) has been attained as

Case 6. If we get P:1-m2,  Q:2-m2, then F(ξ)=scξ, therefore

As long as m→0,F(ξ)=tanξ, and we obtain one of the traveling wave solutions of Eq. (12) as

Case 7. For choices P:1,Q:(2-m2) from table, F is obtained as F(ξ)=csξ, in this way the solution may be expressed as

Moreover, for m→1 from(8), F(ξ)=cschξ, and one of the solitary wave solutions of Eq. (12) can be found as

Case 8. Setting P:1,Q:(2-m2), then F(ξ)=csξ, due to this settings,

Furthermore, for m→0 by using from (9) ,F(ξ)=cotξ, and we attain one of the traveling wave solutions of Eq. (12) as

Case 9. If we take P=(1/4),Q=(1-2m2/2) it may be deducted from table, F(ξ)=nsξ±csξ, therefore

In this case for m→1,F(ξ)=cothξ±cschξ, and one of the solitary wave solutions of Eq. (12) can be shown as

Case 10. Regarding P=1/4, Q=1-2m2/2, then F(ξ)=nsξ±csξ, so

In addition, for m→0,F(ξ)=cscξ±cotξ, and the periodic solution of Eq. (12) can be obtained as

Case 11. Assigning P=(1-m2)/4,Q=(1+m2)/2, then F(ξ)=ncξ±scξ, hence

In the limit case when m→0,F(ξ)=secξ±tanξ, and the periodic solution of Eq. (12) can be written as

Case 12. If we choose P>0, Q<0 from table, F(ξ)=-m2Q/(1+m2)Psn-Q/(1+m2)ξ, so

In the limit case when m→1,F(ξ)=(-m2Q)/([1+m2]P)tanh-Q/(1+m2)ξ, and the solitary wave solution of Eq. (12) can be stated as

Case 13. For choices P<0, Q>0, then F(ξ)=-Q/(2-m2)PdnQ/(2-m2)ξ, hence

In the limit case when m→1, F(ξ)=-Q/([2-m2]P)Q/(2-m2)ξ, and we get one of the solitary wave solutions of Eq. (12) as

Case 14. While P=1, Q=m2+2, then F(ξ)=dnξcnξ/snξ, therefore

In the limit case when m→0,F(ξ)=cosξ/sinξ, and one of the traveling wave solutions of Eq. (12) can be evaluated as

Case 15. When P=-4/m, Q=6m-m2-1, then F(ξ)=mdnξcnξ/(msn2ξ+1), hence

As long as m→1,F(ξ)=msechξsechξ/(mtanhξ tanhξ+1), and we get one of the solitary wave solutions of Eq. (12) as

Case 16. Setting P=4/m, Q:-6m-m2-1, then F(ξ)=mdnξcnξ/(msn2ξ-1), so

When m→1, F(ξ)=msechξsechξ/(mtanhξtanhξ-1), and we obtain one of the solitary wave solutions of Eq. (12) as

Case 17. If we get P=1/4, Q=(1-2m2)/2, then F(ξ)=snξ/(1±cnξ), hence

If m→1, F(ξ)=tanhξ/(1±sechξ), and one of the solitary wave solutions of Eq. (12) can be stated as

Case 18. Supposing P=1/4,Q=(1-2m2)/2, then F(ξ)=snξ/(1±cnξ), therefore

For m→0,F(ξ)=sinξ/(1±cosξ), and the travelling wave solution of Eq. (12) can be obtained as

Case 19. If we get P=(1-m2)/4, Q=(1+m2)/2, then F(ξ)=cnξ/(1±snξ), hence

In the limit case when m→0, F(ξ)=cosξ/(1±sinξ), and the travelling wave solutions of Eq.(12) can be attained as