Application of extended rational trigonometric techniques to investigate solitary wave solutions

In this paper, a variety of novel exact traveling wave solutions are constructed for the (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document}-dimensional Boiti-Leon-Manna-Pempinelli equation via analytical techniques, namely, extended rational sine-cosine method and extended rational sinh-cosh method. The physical meaning of the geometrical structures for some of these solutions is discussed. Obtained solutions are expressed in terms of singular periodic wave, solitary waves, bright solitons, dark solitons, periodic wave and kink wave solutions with specific values of parameters. For the observation of physical activities of the problem, achieved exact solutions are vital. Moreover, to find analytical solutions of the proposed equation many methods have been used but given methodologies are effective, reliable and gave more and novel exact solutions.


Introduction
Investigating soliton wave solutions to nonlinear partial differential equations has long been a major impact in the field of mathematical physics. Traveling wave solutions of NPDEs in the type of soliton solutions have essential significance since they create a strong relation between mathematics and physics. Exact traveling wave solutions are considered best to understand the phenomena of natural sciences. A better deal of applications of NPDEs therefore appealed numerous researchers to look for their exact solutions. Many methods have been applied to find exact solutions of NPDEs such as, generalized exponential rational function Osman and Ghanbari (2018), tanh method Tariq and Akram (2017), the exp(− )-expansion method Sajid and Akram (2018), the extended rational sine-cosine approach and extended rational sinh-cosh approach Akram (2019a, 2019b), and Boiti et al. Boiti et al. (1986) initially derived the above system that is a model for an incompressible fluid where and are the segments of the dimensionless speed. The system has been generally investigated from different view points, for example, the investigation of its Painlevè property, Lie symmetries, its solutions using arbitrary exponential functions, its the conservation law forms, its exact solution using a separation of variable approach etc. By inserting the transformation: = Φ x and = Φ y , this system of equations yields where Φ = Φ(x, y, t) and this equation (1.1) is called as Boiti-Leon-Manna-Pempinelli (BLMP) equation which was derived by Gilson et al. Gilson et al. (1993) during their researched a (2 + 1)-dimensional generalization of the AKNS shallow-water wave equation using the bilinear method. This equation was utilized to depict the (2 + 1)-dimensional interaction of the Riemann wave propagated along the y-axis with a long wave propagated along the x-axis. Many researchers are focusing to extract exact solutions of BLMP equation using various different methods such as, based on the binary Bell polynomials Luo (2001), Wronskian formalism and the Hirota method Delisle and Mosaddeghi (2013); Najafi et al. (2013), the extended homoclinic test approach Tang and Zai (2015), the rational sine-cosine method Arbabi and Najafi (2016) and so on.
The strategy of the paper is summarized as follows: Demarcation of extended rational sine-cosine and extended rational sinh-cosh approaches are presented, in Section 2. In Section 3, application of these methods on the BLMP equation is investigated and graphs of some obtained solutions are drawn in Section 4. Conclusion is given in Section 5. Section 6 represents future recommendations.

Algorithms
Consider the nonlinear partial differential equation (NPDE): where Φ = Φ(x, t) and inserting the following traveling wave transformation where c refers the wave speed, which converts the NPD Eq. (2.1)into an ODE: where ′ denotes the derivative with respect to .

Extended rational sine-cosine method
Step 1. To obtain the solutions of Eq. (2.2), extended rational sine-cosine method asserts the general solution in the form or, where the unknown parameters 0 , 1 , 2 and is the wave number can be determined later.
Step 2. By substituting Eq. (2.3) or (2.4) into Eq. (2.2), polynomials in cos( ) or sin( ) are obtained. Then collecting all coefficients with like powers of cos( ) z or sin( ) z , (where z is a positive integer) and equating them to zero. A set of algebraic equations can be obtained. The resulting equations are solved with the aid of Maple to get the values of unknown constants 0 , 1 , 2 , c and .
Step 3. Substituting the obtained unknown values from Step 2 into Eq. (2.3) or (2.4), the solution of Eq. (2.2) can be found.

Extended rational sinh-cosh method
Step 1. To obtain the solutions of Eq. (2.2), extended rational sinh-cosh method asserts the general solution in the form or, where the unknown parameters 0 , 1 , 2 and refers the wave number can be determined later.
Step 2. By substituting Eq. (2.5) or (2.6) into Eq. (2.2), polynomials in cosh( ) or sinh( ) are obtained. Then collecting all coefficients with like powers of cosh( ) z or sinh( ) z , (where z is a positive integer) and equating them to zero. A set of equations can be obtained. The resulting equations are solved with the aid of Maple to get the values of unknown constants 0 , 1 , 2 , c and .
Step 3. Substituting the obtained unknown values from Step 2 into Eq. (2.5) or (2.6), the solution of Eq. (2.2) can be found.

Exact solutions of the Proposed PDE
The transformation: where 1 , 2 and c are constants, is inserting into Eq. (1.1) and the resulting ODE can be written as Integrating Eq. (3.2) and setting the constant of integration equals to zero which leads

Exact solutions by extended rational sine-cosine method
Suppose that solution of Eq.  (3.5) .

Conclusion
Exact rational trigonometric solutions of the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation have been retrieved by using extended rational sine-cosine and extended rational sinh-cosh methods. Obtained solitary wave solutions are extremely useful in the study of NPDEs in the context of shallow water waves. Some new graphical representations are obtained with the help of these methods. It is found that some of the obtained exact solutions have likely comparable with Arbabi and Najafi (2016). As solutions of Eq.(47) and Eq.(59) in Arbabi and Najafi (2016) are likely similar with our solutions Eqs.
(3.6) and (3.5) respectively. To our knowledge, remaining obtained solutions such as solitary waves, singular periodic wave, bright soltiton, dark soliton, periodic wave and kink wave solutions are new and novel.

Future recommendations
This paper obtained solitary wave solutions to a nonlinear evolution equation that appears in mathematical physics. These solutions are going to be indeed valuable for conducting future research in this field. One novel future aspect is to consider high dimensional equations for examples with perturbation term(s), fractional temporal evolutions will lead to additional interesting results that will be further closer to realistic situations. Seeking analytical solutions for such equations will be a daunting task. It is believed that the proposed methods are powerful, effective, and may play an important role describing the physical features of various nonlinear complex models.
Funding Not applicable.
Code availability The computations involved in the work are done with the help of Maple and Mathematica.

Conflict of interest
The Authors have no conflict of interest.