Apposite solutions to fractional nonlinear Schrödinger-type evolution equations occurring in quantum mechanics

Nonlinear evolution equations of arbitrary order bearing a significantly broad range of capability to illustrate the underlying behavior of naturalistic structures relating to the real world, have become a major source of attraction of scientists and scholars. In quantum mechanics, the nonlinear dynamical system is most reasonably modeled through the Schrödinger-type partial differential equations. In this paper, we discuss the (2+1)-dimensional time-fractional nonlinear Schrödinger equation and the (1+1)-dimensional space–time fractional nonlinear Schrödinger equation for appropriate solutions by means of the recommended enhanced rational [Formula: see text]-expansion technique adopting Cole–Hopf transformation and Riccati equation. The considered equations are turned into ordinary differential equations by implementing a composite wave variable replacement alongside the conformable fractional derivative. Then a successful execution of the proposed method has been made, which brought out supplementary innovative outcomes of the considered equations compared with the existing results found so far. The well-generated solutions are presented graphically in 3D views for numerous wave structures. The high performance of the employed technique shows the acceptability which might provide a new guideline for research hereafter.

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.