Outset of multiple soliton solutions to the nonlinear Schrödinger equation and the coupled Burgers equation

The nonlinear Schrödinger equation and the coupled Burgers equation illustrate the status of quantum particles, shock waves, acoustic transmission and traffic flow. Therefore these equations are physically significant in their own right. In this article, the new auxiliary equation method has been contrivanced in order to rummage exact wave solutions to previously stated nonlinear evolution equations (NLEEs). We have developed ample soliton solutions and have to do with the physical importance of the acquired solutions by setting the specific values of the embodied parameters through portraying figures and deciphered the physical phenomena. It has been established that the executed method is powerful, skilled to examine NLEEs, compatible to computer algebra and provides further general wave solutions. Thus, the investigation of exact solutions to other NLEES through the new auxiliary method is prospective and deserves further research.


Introduction
Intricate phenomena habitually turn into nonlinear evolution equations (NLEEs). Therefore, the study of NLEEs has sustained to attract much attention in the recent years. Amazingly, the nonlinear phenomena can be investigated by procedures which are interesting and rationally elementary and has a significant impact on both theory and phenomenology. The exact solutions to nonlinear evolution equations are the important tool to comprehend the various physical events which govern the actual world nowadays. Exact solutions plays a significant role in the study of nonlinear physical phenomena in many fields, such as, water wave mechanics, meteorology, electromagnetic theory, plasma physics, solid-state physics, chemical kinematics, optical fiber, nonlinear optics, biochemistry, mathematical chemistry, mathematical physics etc. Therefore, in the past several decades, many scientists and researchers raced to discover good and new methods for solving NLEEs which are important to explain essential nonlinear problems. For example of these methods, the Hirotaʼs bilinear transformation method (Zhou and Ma 2017), the Backlund transform method (Arnous et al 2015), the inverse scattering method (Ablowitz and Musslimani 2016), the first integral method (Tascan and Bekir 2010), the exp-function method (Ma et al 2010), the tanh-function method , Abdel et al 2011, the Jacobi elliptic function method (Ma et al 2018), the ¢ ( ) G G -expansion method (Bekir and Guner 2013;Bekir and Cevikel 2009) To date, much work has been done by many researchers on solution of so many physically significant nonlinear evolution equations. In this article, we introduce and execute a method named the new auxiliary equation method for constructing analytical soliton solutions to the nonlinear Schrödinger equation and the coupled Burgers equation which are important NLEEs to comprehend the phenomena: quantum particles, shock waves, acoustic transmission and traffic flow. Bibi et al (2017) implemented the method and established closed form wave solutions to some NLEEs through this method.

The new auxiliary equation method
Suppose, the general nonlinear evolution equation is of the form Here F is a nonlinear function of uʼs and u=u(x, y, z, t) is the wave function to be evaluated.
Step 1: We consider the traveling wave variable The wave variable (2.2) transforms the NLEE (2.1) into an ODE as follows where prime meaning the ordinary derivative with respect to ξ.
Step 2: As per the new auxiliary equation method, the exact solution of (2.3) supposed to be where c c c , ,..., N 0 1 are constants to be calculated, such that ¹ c 0 N and f (ξ) is the solution of the nonlinear equation Step 3: Balancing the nonlinear terms and the highest order derivatives, we find the positive integer N concerning in equation (2.4).

Formulation of the soliton solutions
In this section, the nonlinear Schrödinger equation and the coupled Burgers equation containing parameters are studied to found the traveling wave solutions by making use of the new auxiliary equation method.

The nonlinear Schrödinger equation
Let us consider the NLSE of the form (Biswas and milovic 2010) where F is a real valued algebraic function and it is important to have the smoothness of the complex function Considering the complex plane C as a two-dimensional linear space R 2 , the function In equation (3.1.1), q is dependent variable and x, t are the independent variable, and the subscripts represent the partial derivative of q with respect to that variables. The first term in (3.1.1) represents the time evolution term, while the second term is due to the group velocity dispersion and the third term represents nonlinearity, where a and b be the coefficient of second and third term respectively. Equation (3.1.1) is a nonlinear partial differential equation (PDE) of parabolic type which is not integrable, in general. For simplicity here we use the special case, = ( ) F s s, also known as the Kerr law of nonlinearity. In this article, we consider the subsequent NLSE, The nonlinear Schrödinger equation (NLSE) plays a vital role in various fields of physical, biological and engineering science. It appears in many applied areas, including nonlinear optics (Trikia and Biswasb 2011), plasma physics (Shukla and Eliasson 2010) and protein chemistry (Molkenthin et al 2010). In 1973, Hasegawa and Tappert showed that the NLSE is the appropriate equation to describe nonlinear optical light pulse propagation through optical fibers (Hasegawa and Tappert 1973). We seek the travelling wave solutions of the NLSE given in equation (3.1.3). In order to search the wave solutions to the NLSE with Kerr law nonlinearity, we introduce the following wave transformation: Separating real and imaginary parts, equation (3.1.5) yields Therefore, from (3.1.6), we obtain Balancing the highest order derivative term  u with the highest power nonlinear term u 3 , gives n=1. According to the new auxiliary equation method, from equation (2.4), the solution of equation ( is the solution of the nonlinear equation (2.5). Now, by means of the solution (3.1.10) and (2.5) from equation (3.1.9), we obtain Solving the above algebraic equations with the aid of Maple, yields For the above values of the constants, we attain the subsequent solutions to the NLSE. Case 1: When -< q pr 4 0 2 and ¹ r 0, Making use of (3.1.11) and (2.6), (2.7) into equation (3.1.10), we obtain Case 2: When -> q pr 4 0 2 and ¹ r 0, Putting (3.1.11) and (2.8), (2.9) into equation (3.1.10), we achieve Case 3: When + < ¹ q p r 4 0, 0 2 2 and =r p, Applying (3.1.11) and (2.10), (2.11) into equation (3.1.10), we accomplish Case 4: When + > ¹ q p r 4 0, 0 2 2 and =r p, Exerting (3.1.11) and (2.12), (2.13) into equation (3.1.10), we attain Case 5: When -< q p 4 0 2 2 and r=p, Employing (3.1.11) and (2.14), (2.15) from equation (3.1.10), we ascertain Case 6: When -> q p 4 0 2 2 and r=p, Making use of (3.1.11) and (2.16), (2.17) from equation (3.1.10), we gain Case 7: When = q pr 4 2 , By means of (3.1.11) and (2.18) from equation (3.1.10), we find out 3.1.24 Case 8: When rp<0, q=0 and ¹ r 0, Inserting (3.1.11) and (2.19), (2.20) into equation (3.1.10), we derive Case 9: When q=0 and =p r, Setting (3.1.11) and (2.21) into equation (3.1.10), we secure Case 12: When = -+ ( ) q p r , Inserting (3.1.11) and (2.26) into equation (3.1.10), we secure Case 13: When p=0, By means of (3.1.11) and (2.27) from equation (3.1.10), we ascertain Case 14: When = = ¹ r q p 0, Making use of (3.1.11) and (2.28) from equation (3.1.10), we gain 3.1.32 When r=q=0, and r=0 embedding the values of the constants lead to constant solutions and hence has not been written here, since constant solutions have no physical significance.
It is observe that, by means of the new auxiliary equation method, we obtain abundant closed form traveling wave solutions of the NLSE, which might be useful to analyse various complex phenomena in physical sciences, nonlinear optics, fluid dynamics, mathematical biology, plasma physics and other areas.

The coupled Burgers equation
In this section, we will examine the homogeneous form of the coupled BE equation. We consider the following system of equations (Jawad et al 2010): where α and β are arbitrary constants.
In fluid mechanics, Burgers equation is one of the crucial model equations (Rezazadeh et al 2019). In addition these equations are used to narrate the structure of shock waves travelling in a viscous fluid (Mittal and Rohila, 2018), traffic flow and acoustic transmission.

3.2.51
Whenever r=q=0 and r=0 implanting the values of the constants lead to consistent arrangements and subsequently has not been composed here, since steady arrangements have no physical essentialness.

Results and discussion
In this section, we have interpreted the obtained solutions to the nonlinear Schrödinger equation and the coupled Burgers equation through depicting some three dimensional figures of the attained solutions with the aid of symbolic computation software Mathematica. In this segment, we have represented the some graphs (Figure 1-7  In this section, we analyze the solution of the coupled Burgers equation attained in this study. It is observed that the secured solutions

Conclusion
In this article, the analytical soliton solutions to the nonlinear Schrödinger equation and the coupled Burgers equation have been successfully obtained by executing the new auxiliary equation method with less computational effort. We have established the solitary wave solutions as well as periodic wave solutions, singular periodic wave solutions, kink shape soliton, singular kink shape soliton, bell shape soliton, singular bell shape soliton and compacton wave solution of nonlinear evolution equations by choosing arbitrary values of the free parameters. The solutions are found by trigonometric, hyperbolic, rational, exponential function by using this method. The obtained solutions are quite practical, well suited and useful. This study demonstrated that the new auxiliary equation method is an influential and efficient technique in searching exact solutions for wide class of problems and can also be applied to other kind of complicated nonlinear problems. We might conclude that the suggested method can be prolonged to solve the nonlinear problems that arise in the theory of solitons and other areas.