AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 256019 10.1155/2014/256019 256019 Research Article Elliptic Travelling Wave Solutions to a Generalized Boussinesq Equation El Achab Abdelfattah Khalique Chaudry Masood Department of Mathematics, Faculty of Sciences University of Chouaïb Doukkali, BP 20, El-Jadida Morocco ucd.ac.ma 2014 1722014 2014 06 10 2013 02 01 2014 17 2 2014 2014 Copyright © 2014 Abdelfattah El Achab. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Travelling wave solutions for the generalized Boussinesq wave equation are studied by using the Weierstrass elliptic function method. As a result, some previously known solutions are recovered, and at the same time some new ones are also given, as well as integrable ones.

1. Introduction

It is well known that investigating the exact travelling wave solutions to nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. In order to obtain the exact solutions, a number of methods have been proposed, such as the homogeneous balance method , the hyperbolic function expansion method , Jacobi elliptic function method , F-expansion method , homotopy analysis method [5, 6], the bifurcation theory method of dynamical systems [7, 8], and Weierstrass elliptic function method . Among these methods, Weierstrass elliptic function method is a powerful mathematic tool to solve nonlinear evolution equations. By using this method, many kinds of important nonlinear evolution equations have been solved successfully [10, 11].

To understand the role of nonlinear dispersion in the formation of patterns in liquid drops, Rosenau and Hyman  introduced and studied a family of fully nonlinear KdV equations K ( m , n ): (1) u t + ( u m ) x + ( u n ) x x x = 0 , m > 0 , 1 < n 3 .

A class of solitary waves (which they named compactons) with the special property that, after colliding with other compacton solutions, they reemerge with the same coherent shape was presented. From then on, compacton solutions attracted a lot of interest . You can refer to , for example, for more details about the properties of compacton solutions. Compacton solutions and solitary solutions of other nonlinear evolution equations such as Boussinesq equations and Boussinesq-like B ( m , n ) equations had been extensively investigated by many authors . Recently, Yan  introduced a class of fully Boussinesq equations B ( m , n ) (2) u t t = ( u m ) x x + ( u n ) x x x x = 0 , m , n and presented some of its compacton solutions when m = n . Liu et al.  utilized the integral approach to investigate its compacton solutions. More recently, Zhu [20, 21] studied Boussinesq-like B ( m , n ) equations (3) u t t + ( u m ) x x - ( u n ) x x x x = 0 , m , n > 1 , u t t - ( u m ) x x + ( u n ) x x x x = 0 , m , n > 1 , u t t + ( u 2 n ) x x + ( u 2 n ) x x x x = 0 , n 1 , by using the extended decomposition method.

By making use of the sine-cosine method technique, Wazwaz  obtained various forms of travelling wave solutions for improved Boussinesq equations with positive or negative exponents. It was highlighted in  that the variants and exponents in the equations directly lead to a qualitative change in the physical structures of the obtained solutions. Lai and Wu [24, 25] proposed an approach for constructing asymptotic solutions in a Sobolev space for generalized Boussinesq equations. Using the tanh methods, Wazwaz  obtained the compacton solutions, solitons, solitary pattern solutions, and periodic solutions for the Boussinesq wave equation (4) u t t - u x x + 3 ( u 2 ) x x + a 1 u x x x x and its generalized form (5) u t t - u x x + 3 2 ( u m ) x x + a 1 u x x x x , where a 1 and m ± 1 are constants.

In this paper, we will consider the travelling wave solutions of the following generalized Boussinesq equations (simply called GB ( m , n ) equation): (6) u t t - u x x + a ( u m ) x x + b ( u n ) x x x x .

The objective of this paper is to investigate the travelling wave solutions of (6) systematically, by applying the Weierstrass elliptic function method. It will be shown that some previously known solutions are recovered, and at the same time, some new ones are also given.

The rest of this paper is organized as follows. In Section 2, we first outline the Weierstrass elliptic function method which will be used in the next section. In Section 3, we give some particular travelling wave solutions of (6). Finally, some conclusions are given in Section 4.

2. Weierstrass Elliptic Functions

Let us consider the following nonlinear differential equation: (7) ( d ϕ ( z ) d z ) 2 = a 0 ϕ 4 + 4 a 1 ϕ 3 + 6 a 2 ϕ 2 + 4 a 3 ϕ + a 4 f ( ϕ ) .

As is well known [27, 28], the solutions ϕ of (7) can be expressed in terms of elliptic functions . It reads as (8) ϕ ( z ) = ϕ 0 + f ( ϕ 0 ) 4 [ ( z , g 2 , g 3 ) - ( 1 / 24 ) f ′′ ( ϕ 0 ) ] , where the primes denote differentiation with respect to ϕ and ϕ 0 is a simple root of f ( ϕ ) . The invariants g 2 , g 3 of elliptic functions ( t , g 2 , g 3 ) are related to the coefficients of f ( ϕ ) by  (9) g 2 = a 0 a 4 - 4 a 1 a 3 + 3 a 2 2 , g 3 = a 0 a 2 a 4 + 2 a 1 a 2 a 3 - a 2 3 - a 0 a 3 2 - a 1 2 a 4 .

When g 2 and g 3 are real and the discriminant (10) Δ = g 2 3 - 27 g 3 2 is positive, negative, or zero, we have different behavior of ( t ) . The conditions  (11) Δ 0 or Δ = 0 , g 2 > 0 , g 3 > 0 lead to periodic solutions, whereas the conditions (12) Δ = 0 , g 2 0 , g 3 0 lead to solitary wave solutions.

If Δ = 0 , then ( t , g 2 , g 3 ) degenerates into trigonometric or hyperbolic functions . Thus, periodic solutions according to (8) are determined by (13) ϕ ( z ) = ϕ 0 + f ( ϕ 0 ) × ( 4 [ ( ( 3 2 ) e 1 t ) - e 1 2 - f ′′ ( ϕ 0 ) 24 + 3 2 e 1 csc 2 ( 3 2 e 1 t ) ] ) - 1 , Δ = 0 , g 3 > 0 , and solitary wave solutions by (14) ϕ ( z ) = ϕ 0 + f ( ϕ 0 ) × ( 4 [ e 1 - f ′′ ( ϕ 0 ) 24 + 3 e 1 csc h 2 ( 3 e 1 t ) - f ′′ ( ϕ 0 ) 24 ] ) - 1 , Δ = 0 , g 3 < 0 , where e 1 = | g 3 | 3 in (13) and e 1 = ( 1 / 2 ) | g 3 | 3 in (14).

3. Travelling Wave Solutions of the Generalized Boussinesq Equations

In this section, we consider the travelling wave solutions of (6). Assume that (6) has an exact solution in the form of a travelling wave (15) u ( x , t ) = u ( ξ ) , ξ = μ ( x - c t - x 0 ) , where μ 0 , c 0 , and x 0 are arbitrary constants. Substituting (15) into (6), we get (16) ( c 2 - 1 ) u ′′ + a ( u m ) ′′ + b μ 2 ( u n ) ′′′′ . Integrating (16) twice and letting the constants of integration be zero give rise to (17) ( c 2 - 1 ) u + a ( u m ) + b μ 2 ( u n ) ′′ = 0 . Making transformation, u n = φ , and (17) becomes (18) ( c 2 - 1 ) φ 1 / n + a ( φ m / n ) + b μ 2 φ ′′ = 0 . Multiply (18) by the integrating factor φ on both sides, and integrating it with respect to ξ , we have (19)    ( d φ d ξ ) 2 = 2 b μ 2 [ C 1 - n ( c 2 - 1 ) n + 1 φ 1 + ( 1 / n ) - a n m + n φ 1 + ( m / n ) ] , where C 1 is an integration constant.

Making the transformation φ = ϕ p , p 0 we get (20) ( d ϕ d ξ ) 2 = 2 b μ 2 p 2 × [ C 1 ϕ 2 - 2 p - n ( c 2 - 1 ) n + 1 ϕ 2 + ( ( 1 / n ) - 1 ) p - a n m + n ϕ 2 + ( ( m / n ) - 1 ) p n ( c 2 - 1 ) n + 1 ] . In order to guarantee the integrability of (20), the powers of ϕ have to be integer numbers between 0 and 4, and therefore we have the following parameter conditions:

if C 1 = 0 , n = 1 , then p { - 2 / m , - 1 / ( m - 1 ) , 2 / ( m - 1 ) , 1 / ( m - 1 ) } ;

if C 1 = 0 , n 1 , n = m + 1 , then p { m / ( m - 1 ) , - m / ( m - 1 ) , 2 m / ( m - 1 ) , - 2 m / ( m - 1 ) } ;

if C 1 0 , n = 1 , then we have m = 5 and p = ± 1 / 2 ;

if n = p = 1 , then we have m = 2 or m = 3 .

Next, using the basic results on Weierstrass elliptic functions shown in Section 2, we will analyze the solutions of (20) in the above cases.

3.1. Case 1

(i) If C 1 = 0 , n = 1 , and p = 1 / ( m - 1 ) , (20) becomes (21) ( d ϕ d ξ ) 2 = 2 ( m - 1 ) 2 b μ 2 [ ( 1 - c 2 ) 2 ϕ 2 - a m + 1 ϕ 3 ] = f ( ϕ ) .

It is easy to see that f ( ϕ ) has two roots: ϕ 0 = 0 and ϕ 0 = ( ( 1 - c 2 ) ( m + 1 ) ) / 2 a . From (8), we can obtain the following solution for (21): (22) ϕ = ϕ 0 + 6 ( m - 1 ) ( ( 1 - c 2 ) ϕ 0 - ( 3 a / ( m + 1 ) ) ϕ 0 2 ) 12 b μ 2 ( ξ ) - ( m - 1 ) ( 1 - c 2 - ( 6 a / ( m + 1 ) ) ) , where the invariants are (23) g 2 = ( m - 1 ) 4 ( 1 - c 2 ) 2 12 ( b μ 2 ) 2 , g 3 = - ( m - 1 ) 6 ( 1 - c 2 ) 3 216 ( b μ 2 ) 3 . Then the discriminant Δ = 0 . The root ϕ 0 = 0 gives the trivial solution ϕ = 0 , and the nonzero solution of (21) can be found by taking ϕ 0 = ( ( 1 - c 2 ) ( m + 1 ) ) / 2 a . Hence, from (11), we have the periodic wave solution to (21) (24) ϕ = ( 1 - c 2 ) ( m + 1 ) 2 a sec 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) for ( 1 - c 2 ) / b < 0 . From (12), we have the solitary wave solution (25) ϕ = ( 1 - c 2 ) ( m + 1 ) 2 a sec h 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) for ( 1 - c 2 ) / b > 0 .

Therefore, when ( 1 - c 2 ) / b < 0 , GB ( m , 1 ) (6) has the following periodic wave solution: (26) u ( x , t ) = ( ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ( 1 - c 2 ) ( m + 1 ) 2 a × sec 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ) 1 / ( m - 1 ) . When ( 1 - c 2 ) / b < 0 , GB ( m , 1 ) (6) has the following solitary wave solution: (27) u ( x , t ) = ( sec h 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ( 1 - c 2 ) ( m + 1 ) 2 a × sec h 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ) 1 / ( m - 1 ) .

Remark 1.

When m > 1 , the solutions (26)-(27) that we obtained are completely the same as (22)–(24) that were obtained by Lai .

(ii) If C 1 = 0 , n = 1 , and p = - 1 / ( m - 1 ) , (20) becomes (28)    ( d ϕ d ξ ) 2 = 2 ( m - 1 ) 2 b μ 2 [ ( 1 - c 2 ) 2 ϕ 2 - a m + 1 ϕ ] = f ( ϕ ) .

It is easy to see that f ( ϕ ) has two roots: ϕ 0 = 0 and ϕ 0 = 2 a / ( ( 1 - c 2 ) ( m + 1 ) ) . From (8), we can obtain that the expression for the solutions of (28) is (29) ϕ = ( - 6 a ( m - 1 ) 2 ( m + 1 ) 12 b μ 2 ϕ 0 ( ξ ) + 5 ( m - 1 ) 2 ( 1 - c 2 ) ϕ 0 - 6 a ( m - 1 ) 2 ( m + 1 ) ) × ( 12 b μ 2 ( ξ ) - ( m - 1 ) 2 ( 1 - c 2 ) ) - 1 , where the invariants are given by (23). Taking the root ϕ 0 = 0 into (29), we get (30) ϕ = - 6 a ( m - 1 ) 2 ( m + 1 ) [ 12 b μ 2 ( ξ ) - ( m - 1 ) 2 ( 1 - c 2 ) ] . Since the discriminant Δ = 0 , thus from (11), we have the periodic wave solution to (28) (31) ϕ = 2 a ( 1 - c 2 ) ( m + 1 ) sin 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) b μ 2 ξ ) , for ( 1 - c 2 ) / b < 0 . From (12), we have the solitary wave solution (32) ϕ = - 2 a ( 1 - c 2 ) ( m + 1 ) sinh 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) b μ 2 ξ ) for ( 1 - c 2 ) / b > 0 .

Therefore, when ( 1 - c 2 ) / b < 0 , the GB ( m , 1 ) (6) has the following periodic wave solution: (33) u ( x , t ) = ( × csc 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ( 1 - c 2 ) ( m + 1 ) 2 a × csc 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ) 1 / ( m - 1 ) . When ( 1 - c 2 ) / b > 0 , the GB ( m , 1 ) (6) has the following solitary wave solution: (34) u ( x , t ) = ( ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) - ( 1 - c 2 ) ( m + 1 ) 2 a × csch 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ) 1 / ( m - 1 ) . When substituting the second root ϕ 0 = 2 a / ( ( 1 - c 2 ) ( m + 1 ) ) into (29), we have (35) ϕ = ( [ 12 b μ 2 ( ξ ) - ( m - 1 ) 2 ( 1 - c 2 ) ] 24 a b μ 2 ( ξ ) + 5 ( m + 1 ) ( m - 1 ) 2 ( 1 - c 2 ) 2 - 6 a ( m - 1 ) 2 ( 1 - c 2 ) ) × ( ( m + 1 ) ( 1 - c 2 ) [ 12 b μ 2 ( ξ ) - ( m - 1 ) 2 ( 1 - c 2 ) ] ) - 1 . So from (11), we have the periodic wave solution to (28): (36) ϕ = 2 a ( 1 - c 2 ) ( m + 1 ) cos 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) , for ( 1 - c 2 ) / b < 0 . From (12), we have the solitary wave solution to (28): (37) ϕ = 2 a ( 1 - c 2 ) ( m + 1 ) cosh 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) . Therefore, when ( 1 - c 2 ) / b < 0 , GB ( m , 1 ) (6) has the following periodic wave solution: (38) u ( x , t ) = ( ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) 2 a ( 1 - c 2 ) ( m + 1 ) × cos 2 ( 1 2 - ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ) 1 / ( 1 - m ) . When ( 1 - c 2 ) / b > 0 , the GB ( m , 1 ) (6) has the following solitary wave solution: (39) u ( x , t ) = ( ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) 2 a ( 1 - c 2 ) ( m + 1 ) × cosh 2 ( 1 2 ( m - 1 ) 2 ( 1 - c 2 ) ( b μ 2 ) ξ ) ) 1 / ( 1 - m ) .

Remark 2.

When m > 1 , the solutions (33)-(34) that we obtained are completely the same as (23)–(25) that were obtained by Lai ; when m < 1 and m - 1 , the solutions (38)-(39) that we obtained are similar as the solutions (26)–(28) that were obtained by Lai .

(iii) If C 1 = 0 , n = 1 , and p = 2 / ( m - 1 ) , (20) becomes (40) ( d ϕ d ξ ) 2 = ( m - 1 ) 2 2 b μ 2 [ ( 1 - c 2 ) 2 ϕ 2 - a m + 1 ϕ 4 ] = f ( ϕ ) . According to (8), the solutions of (40) read (41) ϕ = ϕ 0 + ( 3 ( m - 1 ) 2 [ ( 1 - c 2 ) ϕ 0 - 4 a m + 1 ϕ 0 3 ] ) × ( ( ( ( ( 1 - c ^ 2    ) ) / 2 ) - ( 6 a / ( ( m + 1 )    ) ) ϕ 0 2 ) ( ( 1 - c 2 ) 2 - 6 a m + 1 ϕ 0 2 ) 24 b μ 2 ( ξ ) - ( m - 1 ) 2 × ( ( 1 - c 2 ) 2 - 6 a m + 1 ϕ 0 2 ) ) - 1 , where the invariants are (42) g 2 = ( m - 1 ) 4 ( 1 - c 2 ) 2 192 ( b μ 2 ) 2 , g 3 = - ( m - 1 ) 6 ( 1 - c 2 ) 3 ( 24 b μ 2 ) 3 . Taking the root ϕ 0 = 0 into (41), we get the trivial solution ϕ = 0 . However, when taking the root ϕ = ± ( 1 - c 2 ) ( m + 1 ) / 2 a into (41), we can obtain the solutions to (40) (43) ϕ = ± ( 1 - c 2 ) ( m + 1 ) 2 a 48 b μ 2 ( ξ ) - ( m - 1 ) 2 ( 1 - c 2 ) 48 b μ 2 ( ξ ) + 5 ( m - 1 ) 2 ( 1 - c 2 ) . Since the discriminant Δ = 0 , it is easy to see from (11), (12) that the above solutions (43) will generate the same periodic and solitary wave solutions to (6) as (26), (27).

(iv) If C 1 = 0 , n = 1 , and p = - 2 / ( m - 1 ) , (20) becomes (44)    ( d ϕ d ξ ) 2 = ( m - 1 ) 2 2 b μ 2 [ ( 1 - c 2 ) 2 ϕ 2 - a m + 1 ] = f ( ϕ ) . Using the same arguments as the above, we can deduce that it would give exactly the same solutions of (6) as Case 1 (ii).

3.2. Case 2

(i) If C 1 = 0 , n 1 , n = m , and p = - m / ( m - 1 ) , (20) becomes (45) ( d ϕ d ξ ) 2 = 2 ( m - 1 ) 2 b μ 2 m 2 [ m ( 1 - c 2 ) m + 1 ϕ 3 - a 2 ϕ 2 ] = f ( ϕ ) . Using similar arguments to Case 1 (i), we can deduce that when a / b > 0 , the GB ( m , m ) (6) has the following periodic wave solution: (46) u ( x , t ) = ( a ( m + 1 ) 2 m ( 1 - c 2 ) sec 2 ( 1 2 | m - 1 m | a b μ 2 ξ ) ) - 1 / ( m - 1 ) . When a / b < 0 , the GB ( m , m ) (6) has the following solitary wave solution: (47) u ( x , y , t ) = [ a ( m + 1 ) 2 m ( 1 - c 2 ) sec h 2 ( 1 2 | m - 1 m | - a b μ 2 ξ ) ] - 1 / ( m - 1 ) .

Remark 3.

When m < 1 , the solutions (46)-(47) that we obtained are completely the same as (62)–(64) that were obtained by Lai .

(ii) If C 1 = 0 , n 1 , n = m , and p = m / ( m - 1 ) , (20) becomes (48)    ( d ϕ d ξ ) 2 = 2 ( m - 1 ) 2 b μ 2 m 2 [ m ( 1 - c 2 ) m + 1 ϕ - a 2 ϕ 2 ] = f ( ϕ ) . Using similar arguments to Case 1 (ii), we can deduce that when a / b > 0 , the GB ( m , m ) (6) has the following periodic wave solution: (49) u ( x , t ) = [ 2 m ( 1 - c 2 ) a ( m + 1 ) sin 2 ( 1 2 | m - 1 m | a b μ 2 ξ ) ] 1 / ( m - 1 ) . When a / b < 0 , the GB ( m , m ) (6) has the following solitary wave solution: (50) u ( x , t ) = [ 2 m ( 1 - c 2 ) a ( m + 1 ) sinh 2 ( 1 2 | m - 1 m | a b μ 2 ξ ) ] 1 / ( m - 1 ) .

Remark 4.

When m > 1 , solutions (49)-(50) that we obtained are completely the same as (59)–(61) that were obtained by Lai .

(iii) If C 1 = 0 , n 1 , n = m , and p = - 2 m / ( m - 1 ) , (20) becomes (51) ( d ϕ d ξ ) 2 = ( m - 1 ) 2 2 b μ 2 m 2 [ m ( 1 - c 2 ) m + 1 ϕ 4 - a 2 ϕ 2 ] = f ( ϕ ) . Using similar arguments to Case 2 (i), we can deduce that it would give exactly the same solutions of the GB ( m , m ) (6) as those given by (45) and (46).

(iv) If C 1 = 0 , n 1 , n = m , and p = 2 m / ( m - 1 ) , (52) ( d ϕ d ξ ) 2 = ( m - 1 ) 2 2 b μ 2 m 2 [ m ( 1 - c 2 ) m + 1 - a 2 ϕ 2 ] = f ( ϕ ) . Using similar arguments to Case 2 (ii), we can deduce that it would give exactly the same solutions of the GB ( m , m ) (6) as those given by (48) and (49).

3.3. Case 3

(i) If C 1 0 , n = 1 , m = 5 , and p = 1 / 2 , (20) becomes (53) ( d ϕ d ξ ) 2 = 8 b μ 2 [ C 1 ϕ + ( 1 - c 2 ) 2 ϕ 2 - a 6 ϕ 4 ] = f ( ϕ ) . According to (8), the solutions of (52) read (54) ϕ = 3 b μ 2 ϕ 0 ( ξ ) + 5 ( 1 - c 2 ) ϕ 0 - 2 a ϕ 0 3 + 6 C 1 3 b μ 2 ( ξ ) - ( 1 - c 2 ) + 2 a ϕ 0 2 , where the invariants are given by (55) g 2 = 4 ( 1 - c 2 ) 2 3 ( b μ 2 ) 2 , g 3 = - 8 ( 1 - c 2 ) 3 27 ( b μ 2 ) 3 + 16 a C 1 2 3 ( b μ 2 ) 3 . So we can obtain the general expressions for the solutions to the GB ( 5,1 ) (6): (56) u ( x , t ) = [ 3 b μ 2 ϕ 0 ( ξ ) + 5 ( 1 - c 2 ) ϕ 0 - 2 a ϕ 0 3 + 6 C 1 3 b μ 2 ( ξ ) - ( 1 - c 2 ) + 2 a ϕ 0 2 ] 1 / 2 . For example, substituting the simplest root ϕ = 0 of f ( ϕ ) into (56), we get (57) u ( x , t ) = [ 6 C 1 3 b μ 2 ( ξ ) - ( 1 - c 2 ) ] 1 / 2 .

(ii) If C 1 0 , n = 1 , m = 5 , and p = - 1 / 2 , (20) becomes (58) ( d ϕ d ξ ) 2 = 8 b μ 2 [ C 1 ϕ 3 + ( 1 - c 2 ) 2 ϕ 2 - a 6 ] = f ( ϕ ) . Using similar arguments to Case 3 (i), we can get the following general expression for the solutions to GB ( 5,1 ) (6): (59) u ( x , t ) = [ 3 b μ 2 ϕ 0 ( ξ ) + 3 ( 1 - c 2 ) ϕ 0 + 12 C 1 ϕ 0 2 3 b μ 2 ( ξ ) - ( 1 - c 2 ) - 6 C 1 ] - 1 / 2 , where the invariants are given by (55).

3.4. Case 4

(i) If n = 1 , m = 2 , and p = 1 , (20) becomes (60) ( d ϕ d ξ ) 2 = 2 b μ 2 [ C 1 + ( 1 - c 2 ) 2 ϕ 2 - a 3 ϕ 3 ] = f ( ϕ ) . Following the same procedure as mentioned above, we can get the general solutions to GB ( 2,1 ) (6): (61) u ( x , t ) = [ 12 b μ 2 ϕ 0 ( ξ ) + 5 ( 1 - c 2 ) ϕ 0 - 4 a ϕ 0 2 12 b μ 2 ( ξ ) - ( 1 - c 2 ) + 2 a ϕ 0 ] , where ϕ 0 is the real root of f ( ϕ ) = 0 and the invariants are given by (62) g 2 = 3 4 ( 1 - c 2 ) 2 6 ( b μ 2 ) 2 , g 3 = - ( 1 - c 2 ) 3 216 ( b μ 2 ) 3 - a 2 C 1 108 ( b μ 2 ) 3 . (ii) If n = 1 , m = 3 , and p = 1 , (20) becomes (63) ( d ϕ d ξ ) 2 = 2 b μ 2 [ C 1 + ( 1 - c 2 ) 2 ϕ 2 - a 4 ϕ 4 ] = f ( ϕ ) . Likewise, we can get the general solutions to GB ( 3,1 ) (6): (64) u ( x , t ) = [ 12 b μ 2 ϕ 0 ( ξ ) + 5 ( 1 - c 2 ) ϕ 0 - 3 a ϕ 0 3 12 b μ 2 ( ξ ) - ( 1 - c 2 ) + 3 a ϕ 0 2 ] , where ϕ 0 is the real root of f ( ϕ ) = 0 .

4. Conclusion

From the above discussion, we find the traveling wave solutions of the generalized Boussinesq equation GB ( m , n ) , which are expressed by the hyperbolic functions and trigonometric functions, out without the aid of mathematical software. The results show that the Weierstrass function method is a powerful mathematical tool to search for exact solutions to nonlinear differential equations, especially solitary ones. It may be advantageous that this quite general method can lead to free parameters as shown in the solution. We believe that this approach can also be used to solve other nonlinear equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Wang M. L. Solitary wave solutions for variant Boussinesq equations Physics Letters A 1995 199 3-4 169 172 10.1016/0375-9601(95)00092-H MR1322452 Parkes E. J. Duffy B. R. Travelling solitary wave solutions to a compound KdV-Burgers equation Physics Letters A 1997 229 4 217 220 10.1016/S0375-9601(97)00193-X MR1454316 ZBL1043.35521 Fu Z. Liu S. Liu S. Zhao Q. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations Physics Letters A 2001 290 1-2 72 76 10.1016/S0375-9601(01)00644-2 MR1877700 ZBL0977.35094 Wang M. Zhou Y. The periodic wave solutions for the Klein-Gordon-Schrödinger equations Physics Letters A 2003 318 1-2 84 92 10.1016/j.physleta.2003.07.026 MR2020906 ZBL1098.81770 He J.-H. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons and Fractals 2005 26 3 695 700 2-s2.0-18844426016 10.1016/j.chaos.2005.03.006 He J.-H. Homotopy perturbation method for bifurcation of nonlinear problems International Journal of Nonlinear Sciences and Numerical Simulation 2005 6 2 207 208 2-s2.0-17844387391 Li J. Liu Z. Smooth and non-smooth traveling waves in a nonlinearly dispersive equation Applied Mathematical Modelling 2000 25 1 41 56 2-s2.0-0034317577 10.1016/S0307-904X(00)00031-7 Li J. Liu Z. Traveling wave solutions for a class of nonlinear dispersive equations Journal of Chinese Annals of Mathematics B 2002 23 3 397 418 2-s2.0-0036033675 10.1142/S0252959902000365 Schürmann H. W. Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation Physical Review E 1996 54 4 4312 4320 2-s2.0-0343531639 Schürmann H. W. Serov V. S. Nickel J. Superposition in nonlinear wave and evolution equations International Journal of Theoretical Physics 2006 45 6 1057 1073 10.1007/s10773-006-9100-9 MR2254766 ZBL1101.81054 Nickel J. Serov V. S. Schürmann H. W. Some elliptic traveling wave solutions to the Novikov-Veselov equation Progress in Electromagnetics Research 2006 61 323 331 2-s2.0-33947177047 Rosenau P. Hyman J. M. Compactons: solitons with finite wavelength Physical Review Letters 1993 70 5 564 567 2-s2.0-12044253199 10.1103/PhysRevLett.70.564 Liu Z. Li J. Bifurcations of solitary waves and domain wall waves for KdV-like equation with higher order nonlinearity International Journal of Bifurcation and Chaos 2002 12 2 397 407 10.1142/S0218127402004425 MR1894574 ZBL1042.35067 Wazwaz A. M. New solitary-wave special solutions with compact support for the nonlinear dispersive K ( m , n ) equations Chaos, Solitons and Fractals 2002 13 2 321 330 10.1016/S0960-0779(00)00249-6 MR1860771 ZBL1028.35131 Zhang L. Chen L.-Q. Huo X. Peakons and periodic cusp wave solutions in a generalized Camassa-Holm equation Chaos, Solitons and Fractals 2006 30 5 1238 1249 10.1016/j.chaos.2005.08.202 MR2249231 ZBL1142.35591 Li Y. A. Olver P. J. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. I. Compactons and peakons Discrete and Continuous Dynamical Systems 1997 3 3 419 432 10.3934/dcds.1997.3.419 MR1444203 ZBL0949.35118 Li Y. A. Olver P. J. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system. II. Complex analytic behavior and convergence to non-analytic solutions Discrete and Continuous Dynamical Systems 1998 4 1 159 191 MR1485369 ZBL0959.35157 Rosenau P. Nonlinear dispersion and compact structures Physical Review Letters 1994 73 13 1737 1741 10.1103/PhysRevLett.73.1737 MR1294558 ZBL0953.35501 Yan Z. New families of solitons with compact support for Boussinesq-like B ( m , n ) equations with fully nonlinear dispersion Chaos, Solitons and Fractals 2002 14 8 1151 1158 10.1016/S0960-0779(02)00062-0 MR1924519 ZBL1038.35082 Zhu Y. Exact special solutions with solitary patterns for Boussinesq-like B ( m , n ) equations with fully nonlinear dispersion Chaos, Solitons and Fractals 2004 22 1 213 220 10.1016/j.chaos.2003.12.101 MR2058764 ZBL1062.35125 Zhu Y. Lu C. New solitary solutions with compact support for Boussinesq-like B ( 2 n , 2 n ) equations with fully nonlinear dispersion Chaos, Solitons & Fractals 2007 32 2 768 772 10.1016/j.chaos.2005.11.030 MR2280117 ZBL1139.35091 Liu Z. Lin Q. Li Q. Integral approach to compacton solutions of Boussinesq-like B ( m , n ) equation with fully nonlinear dispersion Chaos, Solitons and Fractals 2004 19 5 1071 1081 10.1016/S0960-0779(03)00275-3 MR2013004 ZBL1068.35135 Wazwaz A.-M. Nonlinear variants of the improved Boussinesq equation with compact and noncompact structures Computers & Mathematics with Applications 2005 49 4 565 574 10.1016/j.camwa.2004.07.016 MR2124387 ZBL1070.35043 Lai S. Wu Y. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation Discrete and Continuous Dynamical Systems B 2003 3 3 401 408 10.3934/dcdsb.2003.3.401 MR1974154 ZBL1124.35335 Lai S. Wu Y. H. Yang X. The global solution of an initial boundary value problem for the damped Boussinesq equation Communications on Pure and Applied Analysis 2004 3 2 319 328 10.3934/cpaa.2004.3.319 MR2059324 ZBL1069.35055 Wazwaz A.-M. Compactons and solitary wave solutions for the Boussinesq wave equation and its generalized form Applied Mathematics and Computation 2006 182 1 529 535 10.1016/j.amc.2006.04.014 MR2292062 ZBL1106.65092 Weierstrass K. Mathematische werke V Johnson 1915 New York, NY, USA Whittaker E. T. Watson G. N. A Course of Modern Analysis 1996 Cambridge, UK Cambridge University Press vi+608 MR1424469 Chandrasekharan K. Elliptic Functions 1985 Berlin, Germany Springer xi+189 MR808396 Abramovitz M. Stegun I. A. Handbook of Mathematical Functions 1972 9th New York, NY, USA Dover Lai S. Different physical structures of solutions for a generalized Boussinesq wave equation Journal of Computational and Applied Mathematics 2009 231 1 311 318 10.1016/j.cam.2009.02.025 MR2532672 ZBL1177.35202