DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

1990 ◽  
Vol 04 (05) ◽  
pp. 1003-1037 ◽  
Author(s):  
A.R. Its ◽  
A.G. Izergin ◽  
V.E. Korepin ◽  
N.A. Slavnov
Keyword(s):  
Correlation Function ◽  
Integral Operator ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Nonlinear Schrödinger Equation ◽  
Correlation Functions ◽  
Quantum Correlation ◽  
Schrodinger Equation ◽  
Bose Gas ◽  
Nonlinear Schrodinger Equation

The quantum nonlinear Schrödinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painlève transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrödinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.

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.

New optical soliton solutions of fractional perturbed nonlinear Schrödinger equation in nanofibers

2021 ◽  
Author(s):  
S. Saha Ray ◽  
N. Das
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
New Exact Solutions

In this article, the space-time fractional perturbed nonlinear Schrödinger equation (NLSE) in nanofibers is studied using the improved [Formula: see text] expansion method (ITEM) to explore new exact solutions. The perturbed nonlinear Schrodinger equation is a nonlinear model that occurs in nanofibers. The ITEM is an efficient method to obtain the exact solutions for nonlinear differential equations. With the help of the modified Riemann–Liouville derivative, an equivalent ordinary differential equation has been obtained from the nonlinear fractional differential equation. Several new exact solutions to the fractional perturbed NLSE have been devised using the ITEM, which is the latest proficient method for analyzing nonlinear partial differential models. The proposed method may be applied for searching exact travelling wave solutions of other nonlinear fractional partial differential equations that appear in engineering and physics fields. Furthermore, the obtained soliton solutions are depicted in some 3D graphs to observe the behaviour of these solutions.

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.

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.

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