Analytical studies for the Boiti–Leon–Monna–Pempinelli equations with variable and constant coefficients

1. Introduction

From physics, chemistry i.e. science to engineering, the nonlinear partial differential equations (NPDEs) have major roles to model the processes. Therefore, exact solutions become more important which represents such as rational solutions, periodic solutions, solitary wave solutions, solitons solutions and complexiton solutions [13]. In the literature, various methods are proposed to obtain exact solutions of NPDEs. Some of them are tanh-method [24], Backlund transformation, the inverse scattering theory [4,27], Hirota’s bilinear method [8], the ( G ′ / G ) -expansion method [9] and etc.

For NPDEs, some equation classes are important and KdV equation is one of them, which describes the long wavelength and small amplitude shallow water waves, stratified internal waves and ionacoustic waves in plasmas. Additionally the variable coefficient equations are of great significance to simulate and describe physical phenomena more realistically than their constant-coefficient equations in physics and engineering such as variable-coefficient KdV, KP, and Boussinesq equation [14,15].

In this work, the exact solutions of the generalized ( 2 + 1 ) -dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is considered (1) u x x x y + 3 ( u x u y ) x + g ( x , t ) u y y + s ( y , t ) u t x + h ( x , t ) u t y + m ( y , t ) u x y = 0 where g, s, h, m are smooth functions with respect to x, y, t, so that it is variable coefficient equation. In the case g ( x , t ) = 0 , s ( y , t ) = 0 , h ( x , t ) = 1 , m ( y , t ) = 0 , constant-coefficient BLMP equation is obtained [16]. The BLMP equation is KdV-type equation and it corresponds to model of the incompressible fluid [18]. With another approach is for Eq. (1) that is generalization of the Asymmetric–Nizhnik–Novikov–Veselov (ANNV) equation that is a two-dimensional KdV equation [17]. Additionally, BLMP equation is also known as generalization of the Ablowitz–Kaup–Newell–Segur (AKNS) equation which is model for shallow-water wave equation [9]. Hence, the BLMP equation has various applications in the area of oceanography and atmospheric science [11].

In recent years, scientists have concentrated on obtaining the exact solutions of Eq. (1). To exact-explicit solutions, bilinear form is constructed via Hirota bilinear method, binary Bell polynomials [17,28]. Lax pairs of Eq. (1) is used to obtain the group transformation, while bilinear Backlund transformation is used for modified Clarkson direct method [11,17].

Some exact solution are obtained by using modified Clarkson Kruskal (CK) direct method [12], by deriving a bilinear form via binary Bell polynomials and with Lax pair by linearising the bilinear form. Additionally, solitary wave solutions are obtained by applying applied two parameter group transformation to the Lax pair [17] while some special type soliton solutions are obtained via the extended homoclinic test approach Tang et al. [23], Deng et al. [7]. Another most used approach is the extended F-expansion method Alofi et al. [2] to get soliton solutions and triangular periodic solutions. In the classical way, multilinear variable separation approach, exact solutions are obtained by Delisle et al. [6], Bai et al. [3]. Kaplan [9] is used the transformed rational function method which includes the exp-function method and the extended tanh-method. Also ( G ′ / G ) -expansion method is used to get exact non-travelling wave solutions of BLMP equation. Baskonus et al. [5,26] considered Boiti–Leon–Pempinelli systems and obtained the exact solutions.

In this work, the exact solutions of Eq. (1) for two types i.e. variable-coefficient BLMP equation and constant-coefficient BLMP equation are investigated by the view of the auxiliary equation method. With this view, the method is a generalization of the mentioned methods. In this generalization, second order ordinary linear differential equation, Chebyshev differential equation and Mathiue differential equation. The last two correspond special functions, while the first is generalization of the known methods such as ( G ′ / G ) -expansion method, the tanh-method etc. [24,25].

As a result, we believe also that obtained results in this paper will play important role for the improvement and implementation of the model.

2. Brief of the auxiliary equation method

In recent years, one of the best known methods to obtain the exact solutions of NPDEs is the auxiliary equation method. The method was first developed using the Riccati differential equation. Then various differential equations are considered as an auxiliary equation [10,19,20,22]. In this work, as mentioned,

For this approach, NPDEs are reduced into nonlinear/linear ODE using the appropriate transformation which is generally wave transformation that is also best known Lie group transformation. The solution of NPDE is considered as a finite series u ( ζ ) = ∑ i = 0 N a i z i ( ζ ) where z ( ζ ) is the solution of the auxiliary equation considered above, N is determined by a balancing principle [21] and the second one is determine the parameters a i ( i = 0 , 1 , … , N ) by symbolic computation.

The auxiliary equation plays a key role in finding the exact solutions of NPDEs, because it is an expansion of z ( ζ ) . Because of the reason, it is important to consider an appropriate auxiliary equation with exact solutions.

As a result, the exact solutions of the BLMP equation are substantial to comment on possible behaviors of the models. The proposed exact approaches, requires lesser parameters compared to the previous approach thereby being a more reliable method.

3. Solutions of BLMP equation

The generalized ( 2 + 1 ) -dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation (Eq. (1)) is considered with constant coefficients for g ( x , t ) = 0 , s ( y , t ) = 0 , h ( x , t ) = 1 , m ( y , t ) = 0 (5) u x x x y + 3 ( u x u y ) x + u y t = 0

With the transformation ζ = β x + μ y − t , the reduced equation is obtained: (6) β 3 u ( 4 ) − 6 β 2 u ′ u ″ + u ″ = 0 Using the balancing principle [21] N = 1 and the ansatz is u ( ζ ) = a 0 + a 1 z ( ζ ) and the parameters are obtained as a solution of the algebraic system.

Now, to solve Eq. (6), the auxiliary equation method with the proposed auxiliary equations is considered, respectively.

Case 1 : The auxiliary equation is the differential equation of Chebyshev polynomials z ″ ( ω ) + n 2 z ( ω ) = 0 , ω = cos ζ and its solution z n ( ζ ) = C 1 sin ( n ζ ) + C 2 cos ( n ζ ) and solution of algebraic system is C 1 = 1 − β 3 n 2 + 6 β 2 a 1 n C 2 sin ( n ζ ) 6 β 2 a 1 n cos ( n ζ ) , C 2 = n 1 / 3 5 2 1 / 6 6 a 1 , β = 2 2 / 3 n − 2 / 3 2 so the exact solution is u ( x , y , t ) = a 0 + a 1 ( ( 2 2 / 3 ( 1 + 10 sin ( n ( 2 2 / 3 n − 2 / 3 2 x + μ y − t ) ) ) 12 n − 1 / 3 a 1 cos ( n ( 2 2 / 3 n − 2 / 3 2 x + μ y − t ) ) ) × sin ( n ( 2 2 / 3 n − 2 / 3 2 x + μ y − t ) ) + ( n 1 / 3 5 2 1 / 6 6 a 1 ) cos ( n ( 2 2 / 3 n − 2 / 3 2 x + μ y − t ) ) ) where a 0 , μ, n, a 1 are parameters that can be determined by the initial and boundary conditions. For the special values of parameters, the plot is given by Fig. 1.

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Fig. 1.

The exact solution via Chebyshev differential equation y = 0.5 , a 0 = 1 , a 1 = 1 , n = 2 for different values of μ = − 1 , − 2 , 1 , respectively.

Case 2 : The auxiliary equation is Mathiue differential equation d 2 z ( ξ ) d ξ 2 + ( a − 2 q cos ( ξ ) ) z ( ξ ) = 0 , which has a solution as z ( ξ ) = C 1 Mathieu C ( a , q , ξ ) + C 2 Mathieu S ( a , q , ξ ) and solution of algebraic system is C 1 = − ( − a − 16 q cos ( 2 ζ ) + 64 β 3 q cos ( 2 ζ ) + β 3 a 2 + 32 β 3 a q cos ( 2 ζ ) + 256 β 3 q 2 cos ( 2 ζ ) 2 + 6 β 2 a 1 C 2 Mathieu S Pr ime ( a , − 8 q , ζ ) a + 96 β 2 a 1 C 2 Mathieu S Pr ime ( a , − 8 q , ζ ) q cos ( 2 ζ ) ) / ( 6 β 2 a 1 Mathieu S Pr ime ( a , − 8 q , ζ ) ( a + 16 q cos ( 2 ζ ) ) )

Therefore, the plots of the exact solutions are given in the following Fig. 2.

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Fig. 2.

The exact solution via Mathieu differential equation x = 2 , a 0 = 1 , a 1 = 1 , μ = − 1 , a = 1 , q = 0.5 , C 2 = 1 for different values of β = − 1 , 2 , respectively.

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Fig. 3.

The exact solution via second order differential equation y = 2 , a 0 = 1 , β = 0.5 , C 2 = 1 , C 2 = − 1 for different values of μ = 1 , − 1 , respectively.

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Fig. 4.

The exact solution via Chebyshev differential equation y = 0.5 , a 0 = 1 , μ = − 1 , β = 1 , n = 2 , C 2 = 1 , C 2 = − 1 for different coefficient functions, respectively.

Case 3 : One of the special case of second order ordinary linear differential equation Eq. (2), P ( ζ ) z ″ + ( P ( ζ ) Q ( ζ ) − P ′ ( ζ ) ) z ′ + P 2 ( ζ ) R ( ζ ) z = 0 where the coefficients P ( ζ ) , Q ( ζ ) , R ( ζ ) are functions of ζ and its solution is given by Ahmed & Kalim [1] z ( ζ ) = ( C 1 + C 2 ζ ) P exp ( − ∫ Q 2 d ζ )

In this case, Q ( ζ ) = P ′ ( ζ ) is obtained from algebraic system and P ( ζ ) = ln ( ζ ) is considered.

Therefore, R ( ζ ) = C 3 ln ( ζ ) is obtained and the exact solution of Eq. (2) is z ( ζ ) = ln ( ζ ) ( C 1 + C 2 ζ ) ζ . With the symbolic computation, a 1 = ζ 3 / 2 ( β 3 C 3 − 1 ) ln ( ζ ) 3 β 2 ( − C 1 − C 2 ζ − ln ( ζ ) C 2 ζ + C 1 ln ( ζ ) ) is obtained.

The exact solution is given u ( x , y , t ) = a 0 + ( ( β x + μ y − t ) ln ( β x + μ y − t ) ( β 3 C 3 − 1 ) ( C 1 + C 2 ( β x + μ y − t ) ) ) / ( 3 β 2 ( − C 1 − C 2 ( β x + μ y − t ) − ln ( ( β x + μ y − t ) ) C 2 ( β x + μ y − t ) + C 1 ln ( β x + μ y − t ) ) ) Therefore, the plots of the exact solutions are given in the following Fig. 3.

As it is known generally, the variable-coefficient differential equations are not solved easily. The solutions of Chebyshev differential equation are specific orthogonal polynomials which form complete orthogonal sets in L 2 and are orthogonal especially suited for approximating other functions. Therefore, for the generalized ( 2 + 1 ) -dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation (Eq. (1)), only Chebyshev differential equation is considered to get the solutions.

Following the given procedure above, a 1 = − − β 3 μ n 2 + μ 2 g ( x , t ) + β s ( y , t ) + μ h ( x , t ) + β μ m ( y , t ) 6 β 2 n μ ( C 1 cos ( n ζ ) − C 2 sin ( n ζ ) ) is obtained and the exact solution is u ( x , y , t ) = a 0 − ( ( − β 3 μ n 2 + μ 2 g ( x , t ) + β s ( y , t ) + μ h ( x , t ) + β μ m ( y , t ) ) × ( C 1 sin ( n ( β x + μ y − t ) ) + C 2 cos ( n ( β x + μ y − t ) ) ) ) / ( 6 β 2 n μ ( C 1 cos ( n ( β x + μ y − t ) ) − C 2 sin ( n ( β x + μ y − t ) ) ) ) The plots of the exact solutions are given in the Fig. 4.

5. Conclusion

In this work, the exact solutions of BLMP equation with variable coefficients and constant coefficients have been reached via various types of auxiliary equation method which considers Chebyshev differential equation, Mathieu differential equation and second order linear differential equation as auxiliary equation. Whereas different transformations, Lax pair, Lie group transformation, etc., are seen in the literature, wave transformation is used to reduce BLMP equation in this work. Although the methods discussed are symbolic calculations, they are low cost and suitable for implementation. The obtained results in this paper relate to general physics and should be verified easily because the presence of new mathematical results may have a significant impact on future BLMP equation research.

Funding

No funding was received.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.