Improved General Mapping Deformation Method for Nonlinear Partial Differential Equations in Mathematical Physics

We use the improved general mapping deformationmethod based on the generalized Jacobi elliptic functions expansionmethod to construct some of the generalized Jacobi elliptic solutions for some nonlinear partial differential equations in mathematical physics via the generalized nonlinear Klein-Gordon equation and the classical Boussinesq equations. As a result, some new generalized Jacobi elliptic function-like solutions are obtained by using this method. This method is more powerful to find the exact solutions for nonlinear partial differential equations.


Introduction
The nonlinear partial differential equations play an important role to study many problems in physics and geometry.The effort in finding exact solutions to nonlinear equations is important for the understanding of most nonlinear physical phenomena.For instance, the nonlinear wave phenomena observed in fluid dynamics, plasma and optical fibers are often modeled by the bell-shaped sech solutions and the kinkshaped tanh solutions.
Many effective methods have been presented, such as inverse scattering transform method [1], Bäcklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], variable separation approach [5], various tanh methods [6][7][8][9], homogeneous balance method [10], similarity reductions method [11,12], (  /)-expansion method [13], the reduction mKdV equation method [14], the trifunction method [15,16], the projective Riccati equation method [17], the Weierstrass elliptic function method [18], the Sine-Cosine method [19,20], the Jacobi elliptic function expansion [21,22], the complex hyperbolic function method [23], the truncated Painlevé expansion [24], the -expansion method [25], the rank analysis method [26], the ansatz method [27,28], the exp-function expansion method [29], and the sub-ODE.method [30], Recently, Hong and Lü [31] put a good new method to obtain the exact solutions for the general variable coefficients KdV equation by using the improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method.In this paper, we use the improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method to obtain several new families of the exact solutions for some nonlinear partial differential equations such as the generalized Klein-Gordon equations and the classical Boussinesq equations which are very important in the mathematical physics.

Description of the Improved General Mapping Deformation Method
Suppose we have the following nonlinear PDE: where  = (, ) is an unknown function,  is a polynomial in  = (, ), and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved.
In the following, we give the main steps of a deformation method.

Journal of Applied Mathematics
Step 1.The traveling wave variable where  is a constant, permits us reducing (1) to an ODE for  = () in the following form: where  is a polynomial of  = () and its total derivatives.
Step 2. Firstly, we assume that the solution equation ( 1) has the following form: where  =  −  and   ,   ( = 0, 1, 2, . . ., ), and  are arbitrary constants to be determined later, while () satisfies the following nonlinear first-order differential equation: Step 3. The positive integer "" can be determined by considering the homogeneous balance between the highest order partial derivative and nonlinear terms appearing in (1).
If "" is not positive integer.In order to apply this method when "N" is not positive integer (fraction or negative integer), we make the following transformations.
(1) When  = /,  ̸ = 0 is a fraction in lowest term, we take the following transformation: (2) When  is negative integer, we take the following transformation: and return to determine the value of  again from the new equation.Therefore, we can get that the value of  in (2) is positive integer.

Application of the Improved General Mapping Deformation Method
In this section, we use the proposed method to construct Jacobi elliptic traveling wave solutions for some nonlinear partial differential equations in mathematical physics, namely, the generalized nonlinear Klein-Gordon equation and the classical Boussinesq equations which are very important in the mathematical physics and have been paid attention by many researchers.

Example 1: The Generalized Nonlinear Klein-Gordon
Equations.The generalized nonlinear Klein-Gordon equation [34] is in the following form: where , , , and  are arbitrary constants.These equations play an important role in many scientific applications, such as the solid state physics, the nonlinear optics, and the quantum field theory (see [17,18,24]).Wazwaz [19,35] investigated the nonlinear Klein-Gordon equations and found many types of exact traveling wave solutions including compact solutions, soliton solutions, solitary pattern solutions, and periodic solutions using the tanh-function method.The traveling wave variable (2) permits us to convert equation (10) into the following ODE: Considering the homogeneous balance between the highest order derivative   and the nonlinear term  +1 in (11), we get  = 2/.According to Step 3, we use the following transformation: where () is a new function of .By substituting ( 12) into (11), we get the following new ODE: Determining the balance number  of the new equation ( 13), we get  = 1.Consequently, we have the formal solution of (13) in the following form: where We substitute (14) along with condition (15) into ( 13) and collect all terms with the same power of   ()[  ()]  , ( = 0, 1;  = . . ., −2, −1, 0, 1, 2, . ..).Setting each coefficient of this polynomial to be zero, we get a system of the algebraic equations for  0 ,  1 ,  1 ,  0 ,  1 ,  2 ,  3 ,  4 , and .Also, we substitute ( 8) and ( 9) into (15).Cleaning the denominator and collecting all terms with the same degree of (, ), (, ), and (, ) together, the left-hand side of ( 15) is converted into a polynomial in (, ), (, ), and (, ).Setting each coefficients (, ), (, ), and (, ) of these polynomials to be zero, we derive a system of the algebraic equations for   =   ( = 0, . . ., 4), , , , and .
where , , ,  3 , and  are arbitrary constants.In this case, the rational Jacobi elliptic solution has the following form: Consequently, the exact solution of the generalized nonlinear Klein-Gordon equation (10) takes the following form: where  =  − .In the special case when  = 0, the trigonometric exact solution takes the following form: Case 2.
where , , ,  3 , and  are arbitrary constants.In this case, the rational Jacobi elliptic solution has the following form: Consequently, the exact solution of the generalized nonlinear Klein-Gordon equation (10) takes the following form: where  =  − .In the special case when  = 1, the hyperbolic exact solution takes the following form: Case 3. , where , ,  1 ,  3 , and  are arbitrary constants.In this case, the rational Jacobi elliptic solution has the following form:

𝑑𝑛 (𝜉, 𝑚) ] ]
) Consequently, the exact solution of the generalized nonlinear Klein-Gordon equation ( 10) takes the following form: where  =  − .In the special case when  = 0, the trigonometric exact solution takes the following form: Case 4.
where , , ,  3 , and  are arbitrary constants and  = √ −1.In this case, the rational Jacobi elliptic solution has the following form: Consequently, the exact solution of the generalized nonlinear Klein-Gordon equation ( 10) takes the following form: where  =  − .There are many cases that are omitted for convenience.

Example 2:
The Classical Boussinesq Equations.In this subsection, we consider the classical Boussinesq equations [36] in the following form: The system (31) is integrable and has three Hamiltonian structures [36].Wu and Zhang [37] derive three sets of classical Boussinesq model equations for modeling nonlinear and dispersive long gravity wave traveling in two horizontal directions on shallow water of uniform depth.This system is considered to be one of them.
where , , and  3 are arbitrary constants.In this case, the rational Jacobi elliptic solution has the following form: where  =  − .In the special case when  = 0, the trigonometric exact solution takes the following form: Case 2.
where , , and  3 , are arbitrary constants.In this case, the rational Jacobi elliptic solution has the following form: where  =  − .In the special case when  = 0, the trigonometric exact solution takes the following form: Case 3.
where , , and  1 are arbitrary constants.In this case, the rational Jacobi elliptic solution has the following form: where  =  − .In the special case when  = 1, the hyperbolic exact solution takes the following form: