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

2021 ◽  
pp. 2150470
Author(s):  
Md. Tarikul Islam ◽  
Md. Ali Akbar ◽  
Ozkan Guner ◽  
Ahmet Bekir
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Evolution Equations ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Schrödinger ◽  
Fractional Nonlinear Schrödinger Equation

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.

scholarly journals Non topological 1-soliton solution to resonant nonlinear schrodinger equation with kerr law nonlinearity

2017 ◽  
Vol 5 (1) ◽  
pp. 17
Author(s):  
Salam Subhaschandra Singh
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Soliton Solution ◽  
Schrodinger Equation ◽  
Evolution Equations ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Schrödinger ◽  
Kerr Law ◽  
Kerr Law Nonlinearity

In this paper, using the methods of ansatz, sine-cosine and He’s semi-inverse variation, non-topological 1-soliton solution to Resonant Nonlinear Schrodinger Equation with Kerr law nonlinearity is obtained. The results show that these methods are very effective ones for finding exact solutions to various types of nonlinear evolution equations appearing in the studies of science and engineering.

Construction of dispersive optical solutions of the resonant nonlinear Schrödinger equation using two different methods

2018 ◽  
Vol 32 (33) ◽  
pp. 1850407 ◽  
Author(s):  
E. Tala-Tebue ◽  
Aly R. Seadawy
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Optical Fibers ◽  
Schrodinger Equation ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Schrödinger ◽  
Exponential Function Method

The resonant nonlinear Schrödinger equation is studied in this work with the aid of two methods, namely the exponential rational function method and the modified exponential function method. This equation is used to describe the propagation of optical pulses in nonlinear optical fibers. Being concise and straightforward, these methods are used to build new exact analytical solutions of the model. The solutions obtained are not yet reported in the literature. The methods proposed can be extended to other types of nonlinear evolution equations in mathematical physics.

New types of soliton solutions for space-time fractional cubic nonlinear Schrodinger equation

2021 ◽  
Vol 39 (2) ◽  
pp. 121-131
Author(s):  
Ahmad Neirameh ◽  
Mostafa Eslami ◽  
Mostafa Mehdipoor
Keyword(s):  
Schrödinger Equation ◽  
Differential Equations ◽  
Nonlinear Schrödinger Equation ◽  
Traveling Wave ◽  
Schrodinger Equation ◽  
Space Time ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Schrödinger ◽  
Cubic Nonlinear Schrödinger Equation

New definitions for traveling wave transformation and using of new conformable fractional derivative for converting fractional nonlinear evolution equations into the ordinary differential equations are presented in this study. For this aim we consider the time and space fractional derivatives cubic nonlinear Schrodinger equation. Then by using of the efficient and powerful method the exact traveling wave solutions of this equation are obtained. The new definition introduces a promising tool for solving many space-time fractional partial differential equations.

Radiation solution of the nonlinear Schrödinger equation

1985 ◽  
Vol 63 (5) ◽  
pp. 632-641 ◽  
Author(s):  
R. H. Enns ◽  
S. S. Rangnekar
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Evolution Equations ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Point Of View ◽  
Nonlinear Evolution Equations ◽  
Physical Point ◽  
Nonlinear Schrödinger

The sine-Gordon, sinh-Gordon, modified Korteweg-deVries (KdV), and nonlinear (cubic) Schrödinger equations are four of the most important (from a physical point of view) nonlinear evolution equations that fit into the Ablowitz–Kaup–Newell–Segur (AKNS) inverse scattering framework. Historically, the soliton solutions of these equations have been exhaustively studied in the literature, the radiation solutions being almost entirely neglected. Using an expansion approach (expanding the reflection coefficient in powers of the area A of the input potential), we have previously studied the complete spatial and temporal evolution of the radiation solutions of the first three equations cited above. In this paper we demonstrate, using an illustrative example, that the expansion approach also works for the nonlinear Schrödinger equation. The qualitative features of the radiation solution thus obtained are easily understood in the physical context of the self-defocusing of an intense optical beam. Making use of numerical simulation, we show that the features persist as A is increased to very large values. The solution is also found to be analytically consistent with the asymptotic (t → ∞) form quoted in the literature for a general input profile.

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

2021 ◽  
Author(s):  
Md. Tarikul Islam ◽  
Francisco Gomez ◽  
Md. Ali Akbar
Keyword(s):  
Schrödinger Equation ◽  
Differential Equations ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Kdv Equation ◽  
Nonlinear Schrodinger Equation ◽  
Auxiliary Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Fractional Nonlinear Schrödinger Equation

Abstract 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.

Sign in / Sign up

Export Citation Format

Share Document