scholarly journals Linear System of Differential Equations with a Quadratic Invariant as the Schrödinger Equation

2021 ◽  
Author(s):  
V. V. Kozlov
Keyword(s):  
Linear System ◽  
Schrödinger Equation ◽  
Differential Equations ◽  
Schrodinger Equation ◽  
Real Hilbert Space ◽  
Complex Structure ◽  
Complete Orthonormal System ◽  
Basic Principles ◽  
Linear Partial Differential Equations ◽  
Quadratic Invariant

Abstract Linear systems of differential equations with an invariant in the form of a positive definite quadratic form in a real Hilbert space are considered. It is assumed that the system has a simple spectrum and the eigenvectors form a complete orthonormal system. Under these assumptions, the linear system can be represented in the form of the Schrödinger equation by introducing a suitable complex structure. As an example, we present such a representation for the Maxwell equations without currents. In view of these observations, the dynamics defined by some linear partial differential equations can be treated in terms of the basic principles and methods of quantum mechanics.

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.

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.

MORE ON PARASUPERSYMMETRIES OF THE SCHRÖDINGER EQUATION

1993 ◽  
Vol 08 (05) ◽  
pp. 435-444 ◽  
Author(s):  
J. BECKERS ◽  
N. DEBERGH ◽  
A.G. NIKITIN
Keyword(s):  
Schrödinger Equation ◽  
Differential Equations ◽  
Schrodinger Equation ◽  
Interesting Case ◽  
Second Order ◽  
Schrödinger Equations ◽  
One Dimensional ◽  
Physical Systems ◽  
Systems Of Differential Equations ◽  
General Systems

One-dimensional spatial physical systems described by Schrödinger equations with time-independent interactions admit nth order parasupersymmetries. The general systems of differential equations for the parasupersymmetric operators are obtained and superposed with previous supersymmetric results. The interesting case of second order parasupersymmetries is completely solved.

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.

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