Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method

3.1 Zoomeron Equation

In this subsection, we implement the auxiliary equation method to solve the Zoomeron equation in the form

where u(x, y, t) is the amplitude of the relative wave mood. This equation is one of incognito evolution equation. The equation was introduced by Calogero and Degasperis [29]. Using the wave wariable

Equating (5) is carried to an ODE

where the prime denotes the derivation with respect to ξ and r is the integration constant. Balancing the highest-order derivative term U″ with the nonlinear term U³ of (7) yields N=1.

Thus, we have

Here, a₀, a₁, and a₂ are constants to be determined, while F(ξ) is an unknown function to be determined. From (8), it is easy to see that

and

Then we substitute (8)–(10) into (7) and collect all the terms with the same power of Fⁱ(i=0, 1, …, 6). Equating each coefficient to zero yields a set of the following algebraic equations:

Solving the set of algebraic equations by Maple, we have the following results:

Substituting the values of a and c from (12) into (1), we obtain an ODE. Solution of this ODE for F(ξ) yields

Finally we substitute (13) into (8) and we get the exact solution of (5):

where ξ = x + dy − wt, a = − r2d(w2 − 1) and c₁ is an arbitrary constant.

Note comparing our solutions with [30–32], it can be seen that by choosing suitable values for the parameters, similiar solutions can be verified.

3.2 Coupled Higgs Equation

We use the auxiliary method to obtain new and general exact solutions of the coupled Higgs equation

Tajiri [33] obtained N-soliton solutions to the system (15). Zhao [34] and Bekir [35] constructed more general travelling wave solutions of the equation system (15).

Using the travelling wave transformation

equation system (15) is reduced to the following ODE system:

Here the prime denotes derivation with respect to ξ. Integrating the second equation in the system (17) and neglecting the constant of integration we find

then substituting (18) into the first equation of the system and integrating we find

where the prime denotes the derivation with respect to ξ. Balancing the highest-order derivative term U″ and the nonlinear term U³ from (19) yields 3N=N+2, which gives N=1. Hence, U(ξ) takes the form (8). Substituting (8)–(10) into (19), and equating the coefficients of Fⁱ(i=0, 1, …, 6) to zero, we separately obtain

Solving the algebraic equation system by Maple, we get

and

Substituting (22) into (4) and solving the ODE, we have

Then we substitute (23) into (8) and we get the exact travelling wave solutions of equation system (15):

and from (18)

where a = p2 − r22(w2 − 1), ξ = x + wt and c₁ is an arbitrary constant.

Note that our solutions are more extensive than the given ones [35–37]. To our knowledge, other solutions of (15) that we acquired in this article are new and are not trackable in the previous literature.

3.3 Equal Width Wave Equation

We finally consider the equal width wave (EW) equation in the form [38]

applying the transformation

to (27) yields

where the prime denotes the derivative with respect to ξ. Balancing the highest-order derivative term U″ and the nonlinear term U² of (29) yields N=2. Consequently, (3) takes the form

Here, a_i (i=0, 1, …, 4) are constants to be determined, while F(ξ) is an unknown function to be determined. From (30) along with the auxiliary equation (4), it is easy to see that

and

We substitute (29)–(31) and collect all the terms with the same power of Fⁱ (i=0, 1, …, 8). Equating each coefficient to zero yields a set of the following algebraic equations:

Solving the equation system (32), we have the following results:

Case 1: When

and

Substituting (34) into (4), we obtain an ODE. Solution of this equation yields:

Then, we substitute (35) into (30) and we find the exact solutions of (27) as follows:

U(ξ) = 24bdc1(coshξ − sinhξ) + 2b − 48db2(c1(coshξ − sinhξ) + 2b)2,

where c₁ is an arbitrary constant and ξ=x–dt.

Case 2: When

and

Substituting (37) into (4), we obtain an ODE. Solution of this equation yields

and

Then, we substitute (38) into (30) and we find the exact solutions of (27) as follows:

U(ξ) = 2d + 24dbc1(cosξ − isinξ) − 2b + 48db2(c1(cosξ − isinξ) + 2b)2.

Also by substituting (39) into (30), we get the exact solutions of (27) as follows:

U(ξ) = 2d − 24dbc1(cosξ2 − isinξ2)2(c1(cosξ − isinξ) − 1) + 48db2c1(cosξ2 − isinξ2)4(c1(cosξ − isinξ) − 1)2

where c₁ is an arbitrary constant and ξ=x–dt.

Case 3: When

and

Substituting (41) into (4), we obtain an ODE. Solution of this equation yields

Then, we substitute (42) into (30) and we find the exact solutions of (27) as follows:

U(ξ) = 2d − 3d(tan(− ξ2 + c1)2 + 1)tan((− ξ2) + c1)2,

where c₁ is an arbitrary constant and ξ=x–dt.

Note that our solutions are new and more extensive than the given ones in Raslan [39]. When the parameters are given special values, the solitary waves are derived from the travelling waves.

The transformed rational function method [17] paves a way of handling the solution process of nonlinear equations, unifying the tanh-function-type methods, exp-function method, mapping method, homogeneous balance method, and F-expansion-type methods. Also, its key point is to find rational solutions not only to variable-coefficient ODEs but also constant coefficient ones transformed from given nonlinear PDEs. The method given by Ma and Fuchssteiner [22] is a kind of combination of the direct method and the ansatz method. In this study, we obtain various travelling wave solutions including bright and dark solitons, periodic solutions, and exponential solutions.This method helps us find solutions to nonlinear equations through quadratures. In this article, for special cases of a, b, and c, we obtain exponential and trigonometric solutions of referred equations. For example, taking c=0 in (4), we get exponential solutions. The newly proposed technique has many advantages; it is straightforward and concise.