Diverse Soliton Structures for Fractional Nonlinear Schrodinger Equation, KdV Equation and WBBM Equation Adopting a New Technique

: Nonlinear fractional order partial differential equations standing for the numerous dynamical systems relating to nature world are supposed to by unraveled for depicting complex physical phenomena. In this exploration, we concentrate to disentangle the space and time fractional nonlinear Schrodinger equation, Korteweg-De Vries (KdV) equation and the Wazwaz-Benjamin-Bona-Mahony (WBBM) equation bearing the noteworthy significance in accordance to their respective position. A composite wave variable transformation with the assistance of conformable fractional derivative transmutes the declared equations to ordinary differential equations. A successful implementation of the proposed improved auxiliary equation technique collects enormous wave solutions in the form of exponential, rational, trigonometric and hyperbolic functions. The found solutions involving many free parameters under consideration of particular values are figured out which appeared in different shape as kink type, anti-kink type, singular kink type, bell shape, anti-bell shape, singular bell shape, cuspon, peakon, periodic etc. The performance of the proposed scheme shows its potentiality through construction of fresh and further general exact traveling wave solutions of three nonlinear equations. A comparison of the achieved outcomes in this investigation with the results found in the literature ensures the diversity and novelty of ours. Consequently, the improved auxiliary equation technique stands as efficient and concise tool which deserves further use to unravel any other nonlinear evolution equations arise in various physical sciences like applied mathematics, mathematical physics and engineering.


Introduction
The nature is full of nonlinear wonders which has become an important matter of fact to be disclosed by the scholars and researchers day by day. These phenomena are modeled through the nonlinear partial differential equations which play significant roles to depict the underlying mechanisms of numerous complex physical phenomena relating to real world problems. These equations formulate noteworthy phenomena arise in applied sciences such as applied mathematics, mathematical physics, solid state physics, fluid mechanics, biology, solid state biology, chemistry, electric control theory, system identification, chaotic dynamical behavior of dynamical systems, signal processing, quasi-chaotic dynamical systems, food engineering, hydrodynamics, economics, finance and diffusion models [1][2][3][4][5][6]. Consequently, mathematicians and physicists pay deep attention to unravel the above models through the formulation of their solutions in approximate and appropriate form by computational and analytical techniques. A few methods to investigate nonlinear partial differential equations are available in ref. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
But no technique is unique to examine exact traveling wave solutions to nonlinear evolution equations. That is why, experts are hunting new methods incessantly. Khater et al. (2017) have introduced Khater method and found abundant elliptic and solitary wave solutions of three nonlinear equations [22]. This method has been applied by Bibi et al. (2017) to nonlinear Sharma Tasso-Olever (STO) equation of fractional order and found exact analytic solutions [23]. Later, new auxiliary equation method, modified auxiliary equation method and generalized Khater method mostly relevant to Khater method have been presented for analytic wave solutions to different nonlinear partial differential equations [24][25][26][27][28][29][30][31]. wave solutions [40]; the local variational iteration method and the local fractional series expansion method has been adopted by Jasim and Baleanu (2019) to study the fractional KdV equation [41]; Liu and Zhang (2018) have investigated the same equation by improved ( ′ / )-expansion method and found exact analytic solutions [42]; Jacobi elliptic function expansion method has been assumed to examine the KdV equation by Dascioglu et al. (2017) [43]. (c) Theorem 2: Let an -differentiable and also differentiable function be Φ( ) for 0 < ≤ 1 in which another function Ψ( ) is also supposed to be differentiable, then (Φ( ). Ψ( )) = 1− Ψ′( )Φ′(Ψ( )).
This paper makes available abundant distinct and novel exact traveling wave solutions. Finally, we bring out a comparable study of the gained outcomes with the existing results in the literature which ensures the newness and generality of our well-furnished solutions. This work might encourage the scholars for further study and create a landmark in the research area.

Elucidation of the proposed improved auxiliary equation technique
We take into account the following type non-integer order nonlinear evolution equation: The procedures of the proposed technique are as follows: The solution of Eq. (2.1) is supposed to be in the form where ′ and ′ are unknown parameters; is determined by applying homogeneous balance to Eq. Now, we formulate the closed form wave solutions of the following space and time fractional nonlinear partial differential equations:

The nonlinear Schrodinger equation of fractional order
Consider the space and time fractional nonlinear Schrodinger equation where and are non-zero real constants; is the spatial variable and represents time [35].
This equation occurs in plasma physics, non-linear optics and superconductivity. We introduce the fractional composite transformation as follows: where , and are free parameters to be determined later. Eq. (3.1) with the aid of Eq. (3.2) reduces to the ODE The imaginary part yields = 1 and hence the real part becomes Homogenous balance technique due to ′′ and 3 forces Eq. (…) to be Eq. (3.5) alongside its necessary derivatives and Eq. (2.5) reduces Eq. (3.4) to a polynomial in ( ) . Assigning each coefficient to zero and solving by computational software Maple provides the following results: The above cases produce much more solutions to the fractional nonlinear Schrodinger equation which all seem to be absurd to record here. We think reasonable to deal with only one case and thereupon considering case 1 and the solutions of Eq. (2.5) the wave solutions are given as follows: When 2 − 4 < 0 and ≠ 0, According to the condition 2 − 4 > 0 and ≠ 0, For 2 + 4 2 < 0, ≠ 0 and = − , If 2 + 4 2 > 0, ≠ 0 and = − , Once 2 − 4 2 < 0 and = , Making a comparable study of the found solutions to those existing ever it is claimed that this study ensures the diversity and novelty of our results [32][33][34][35][36][37][38].

The space and time fractional nonlinear KdV equation
The  We hunt the earlier recorded results for the nonlinear KdV equation and conclude that our gained outcomes are different and novel [39][40][41][42][43].

The time and space fractional nonlinear WBBM equation
The integer order WBBM equation has been derived by Wazwaz in 2017 which is a modified form of BBM equation and introduced as the wide-ranging model for scientific phenomena [47]. Thereafter, Seadawy et al. (2019) has considered the equation to be fractional order [45].
This equation is given as follows [44]: . The above cases harvest a lot of solutions which all are not recorded here for simplicity.
For the assumption 2 + 4 2 < 0, ≠ 0 and = − , The above obtained solutions to the WBBM equation are compared with those available in the earlier study and claimed to be recorded in the literature for the first time [44,45].

Graphical appearances of found solutions
The achieved solutions of the fractional nonlinear Schrodinger equation, KdV equation and WBBM equation are figured out for their physical appearances. Different shapes of solitons like kink shape, anti-kink shape, bell shape, anti-bell shape, peakon, compacton and periodic are brought out graphically. A few graphs are given below:

Conclusions
Our This method is efficient, concise and inventive which might be put forwarded for further use to look for soliton and other types solutions of any nonlinear fractional models relating to physical sciences. The obtained solutions cover much more free constants and claim to be new, interesting and significant which might be useful to depict the underlying structures of intricate behavior of nature world. So far, we hunt the literature the gained outcomes might newly be visible in the research field and inspire the scientists and scholars to advance the related work in future.

Author Statement
Md. Tarikul Islam