The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics

and Applied Analysis 3 3. Applications In this section, we will apply the proposed method described in Section 2 to find the exact traveling wave solutions of the nonlinear equations (1)–(3). Example 1 (the nonlinear Pochhammer-Chree equation (1)). In this example, we find the exact solutions of (1). To this end, we see that the traveling wave variable (5) permits us to convert (1) into the following ODE: c 2 u 󸀠󸀠 − c 2 u 󸀠󸀠󸀠󸀠 − (αu + βu n+1 + γu 2n+1 ) 󸀠󸀠 = 0. (17) Integrating (17) twice with respect to ξ and vanishing the constants of integration, we get (c 2 − α) u − c 2 u 󸀠󸀠 − βu n+1 − γu 2n+1 = 0. (18) By balancing u󸀠󸀠 with u2n+1 in (18) we getm = 1/n. According to Step 3, we use the transformation u (ξ) = V1/n (ξ) , (19) where V(ξ) is a new function of ξ. Substituting (19) into (18), we get the new ODE (c 2 − α) n 2V2 − c2nVV󸀠󸀠 − c 2 (1 − n) (V󸀠) 2 − βn 2V3 − γn2V4 = 0. (20) Balancing VV󸀠󸀠 with V4 in (20), we get m = 1. Consequently, we get V (ξ) = A 0 + A 1 σ (ξ) + B 1 τ (ξ) , (21) where A 0 , A 1 , and B 1 are constants to be determined later. Substituting (21) into (20) and using (8)-(9) with ε = −1, the left-hand side of (20) becomes a polynomial in σ and τ. Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations: σ 4 : − n 2 γA 4 1 R 2 + R (c 2 nA 2 1 − c 2 nμ 2 A 2 1 − c 2 μ 2 A 2 1 + c 2 A 2 1 − 6γn 2 μ 2 A 2 1 B 2 1 + 6γn 2 A 2 1 B 2 1 ) + 2c 2 nμ 2 B 2 1 − c 2 nμ 4 B 2 1 − c 2 nB 2 1 − c 2 μ 4 B 2 1 + 2c 2 μ 2 B 2 1 − c 2 B 2 1 − γn 2 μ 4 B 4 1 + 2γn 2 μ 2 B 4 1 − γn 2 B 4 1 = 0, σ 3 : 2c 2 μ 3 B 2 1 − 2c 2 μB 2 1 + 2c 2 nA 0 A 1 + 4n 2 γμ 3 B 4 1 + 2Rc 2 μA 2 1 − Rn 2 βA 3 1 − 3c 2 nμB 2 1 − 4n 2 γμB 4 1 + 3n 2 βA 1 B 2 1 + 3c 2 nμ 3 B 2 1 + 12n 2 γA 0 A 1 B 2 1 − 3n 2 βμ 2 A 1 B 2 1 + Rc 2 nμA 2 1 − 4Rn 2 γA 0 A 3 1 − 2c 2 nμ 2 A 0 A 1 + 12Rn 2 γμA 2 1 B 2 1 − 12n 2 γμ 2 A 0 A 1 B 2 1 = 0, σ 3 τ : − 2c 2 nμ 2 A 1 B 1 + 2c 2 nA 1 B 1 − 2c 2 μ 2 A 1 B 1 + 2c 2 A 1 B 1 − 4γn 2 μ 2 A 1 B 3 1 − 4Rγn 2 A 3 1 B 1 + 4γn 2 A 1 B 3 1 = 0, σ 2 : R 2 (−c 2 A 2 1 − 6γn 2 A 2 1 B 2 1 ) + R (3c 2 nμA 0 A 1 + 2c 2 nB 2 1 − c 2 μ 2 B 2 1 − 6γn 2 μ 2 B 4 1 + 24γn 2 μA 0 A 1 B 2 1 + 6βn 2 μA 1 B 2 1 − 6γn 2 A 2 0 A 2 1 − 3βn 2 A 0 A 2 1 − αn 2 A 2 1 + 2γn 2 B 4 1 + c 2 n 2 A 2 1 − 3c 2 nμ 2 B 2 1 ) + c 2 n 2 μ 2 B 2 1 − c 2 n 2 B 2 1 − 6γn 2 μ 2 A 2 0 B 2 1 − 3βn 2 μ 2 A 0 B 2 1 − αn 2 μ 2 B 2 1 + 6γn 2 A 2 0 B 2 1 + 3βn 2 A 0 B 2 1 + αn 2 B 2 1 = 0, σ 2 τ : n 2 βB 3 1 + 2c 2 nA 0 B 1 − n 2 βμ 2 B 3 1 + 4n 2 γA 0 B 3 1 + 2Rc 2 μA 1 B 1 − 4n 2 γμ 2 A 0 B 3 1 − 3Rn 2 βA 2 1 B 1 − 2c 2 nμ 2 A 0 B 1 − 12Rn 2 γA 0 A 2 1 B 1 + 2Rc 2 nμA 1 B 1 + 8Rn 2 γμA 1 B 3 1 = 0, σ : 2c 2 n 2 A 0 A 1 − 2n 2 αA 0 A 1 + 2n 2 αμB 2 1 − 3n 2 βA 2 0 A 1 − 4n 2 γA 3 0 A 1 − 2c 2 n 2 μB 2 1 − Rc 2 nA 0 A 1 + 12n 2 γμA 2

In this paper, we will use the generalized projective Riccati equations method to construct exact solutions for the following three nonlinear evolution equations with higherorder nonlinear terms: (i) the nonlinear Pochhammer-Chree equation [41]: where , , and  are constants and  < 0, (ii) the nonlinear Burgers equation [42]: where  and  are constants; (iii) the nonlinear generalized Zakharov-Kuznetsov equation [43]: where , , and  are nonzero real constants.Zuo [32] has applied the extended (  /)-expansion method and determined the exact solutions of (1), and Hayek [33] has found the exact solutions of (2) using another form of the extended (  /)-expansion method, while Zhang [44] 2 Abstract and Applied Analysis has discussed (3) using an algebraic method to find some of its exact solutions.The rest of this paper is organized as follows.In Section 2, we give the description of the generalized projective Riccati equations method.In Section 3, we apply this method to solve (1)-(3).In Section 4, physical explanations of some obtained results are obtained.In Section 5, some conclusions are given.

Description of the Generalized Projective Riccati Equations Method
Consider we have the following NPDE: where  is a polynomial in (, ) and its partial derivatives, in which the highest-order derivatives and nonlinear terms are involved.In the following, we give the main steps of this method.
Step 1.We use the wave transformation where  is a constant, to reduce (4) to the following ODE: where  is a polynomial in () and its total derivatives, such that  = /.
Step 2. We assume that (6) has the formal solution where  0 ,   , and   are constants to be determined later.The functions () and () satisfy the ODEs where where  and  are nonzero constants.
If  =  = 0, (6) has the formal solution where () satisfies the ODE Step 3. We determine the positive integer  in (7) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in (6).In some nonlinear equations the balance number  is not a positive integer.In this case, we make the following transformations.(a) When  = /, where / is a fraction in the lowest terms, we let and then we substitute ( 12) into (6) to get a new equation in the new function V() with a positive integer balance number.(b) When  is a negative number, we let and then we substitute ( 13) into (6) to get a new equation in the new function V() with a positive integer balance number.
Setting each coefficient to zero yields a set of algebraic equations which can be solved to find the values of  0 ,   ,   , , , and .
Step 5.It is well known [24] that (8) admits the following solutions.
where  is nonzero constant.
Step 6. Substituting the values of  0 ,   ,   , , , and  as well as the solutions ( 14)-( 16) into (7), we obtain the exact solutions of (4).We close this section with the remark that without loss of generality we take  = −1 (similarly the case  = 1 can be done which is omitted here for simplicity).

Applications
In this section, we will apply the proposed method described in Section 2 to find the exact traveling wave solutions of the nonlinear equations ( 1
In this example, we find the exact solutions of (1).To this end, we see that the traveling wave variable (5) permits us to convert (1) into the following ODE: Integrating (17) twice with respect to  and vanishing the constants of integration, we get By balancing   with  2+1 in ( 18) we get  = 1/.According to Step 3, we use the transformation where V() is a new function of .Substituting ( 19) into ( 18), we get the new ODE Balancing VV  with V 4 in (20), we get  = 1.Consequently, we get where  0 ,  1 , and  1 are constants to be determined later.Substituting ( 21) into (20) and using ( 8)-( 9) with  = −1, the left-hand side of (20) becomes a polynomial in  and .Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations: On solving the above algebraic equations using the Maple or Mathematica, we get the following results.
Example 2 (the nonlinear Burgers equation ( 2)).In this example, we study the Burgers equation with power-law nonlinearity (2).To this end, we see that the traveling wave variable (4) permits us to convert (2) into the following ODE: Integrating (32) once with respect  and setting the constant of integration to be zero yield By balancing   with   in (33) we get  = 1/( − 1).

According to
Step 3, we use the transformation where V() is a new function of .Substituting (34) into (33), we get the new ODE Balancing V  with V 2 in (35), we get  = 1.Consequently, we get where  0 ,  1 , and  1 are constants to be determined later.Substituting (36) into (35) and using ( 8)-( 9) with  = −1, the left-hand side of ( 35) becomes a polynomial in  and .
Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations: On solving the above algebraic equations using the Maple or Mathematica, we get the following results.
Example 3 (the nonlinear generalized Zakharov-Kuznetsov equation ( 3)).In this example, we study the generalized Zakharov-Kuznetsov equation with power-law nonlinearity (3).To this end, we use the traveling wave variable where  and  are nonzero constants, to reduce (3) to the following ODE: By balancing   with  2   in (45) we get  = 1/.

According to
Step 3, we use the transformation where V() is a new function of .Substituting ( 46) into (45), we get the new ODE Balancing V 2 V  with V 4 V  in (47), we get  = 1.Consequently, we get where  0 ,  1 , and  1 are constants to be determined later.Substituting (48) into (47) and using ( 8)-( 9) with  = −1, the left-hand side of (47) becomes a polynomial in  and .Setting the coefficients of this polynomial to be zero yields a system of algebraic equations in  0 ,  1 ,  1 , , and , which can be solved using the Maple or Mathematica; we get the following results.

Physical Explanations of Some Obtained Solutions
In this section, we have presented some graphs of the obtained solutions constructed by taking suitable values of involved unknown parameters to visualize the underlying mechanism of the original equation.Using mathematical software Maple, three-dimensional plots of some obtained exact solutions have been shown in Figures

Conclusions
The generalized projective Riccati equations method is used in this paper to obtain some new exact solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation, and the generalized Zakharov-Kuznetsov equation.On comparing our results in this paper with the well-known results obtained in [32,33,[41][42][43][44] we deduce that our results are different and new and are not published elsewhere.The proposed method of this paper is effective and can be applied to many other nonlinear equations in mathematical physics.