Exact solutions of the one-dimensional Russo-Smereka kinetic equation

2000 ◽  
Vol 41 (4) ◽  
pp. 593-603 ◽  
Author(s):  
A. A. Chesnokov
Keyword(s):  
Kinetic Equation ◽  
Exact Solutions ◽  
One Dimensional ◽  
The One

scholarly journals Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator

2017 ◽  
Vol 2017 ◽  
pp. 1-14
Author(s):  
Augusto Beléndez ◽  
Enrique Arribas ◽  
Tarsicio Beléndez ◽  
Carolina Pascual ◽  
Encarnación Gimeno ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Elliptic Integral ◽  
First Integral ◽  
Jacobi Elliptic Function ◽  
Complete Elliptic Integral ◽  
One Dimensional ◽  
The Common ◽  
The One ◽  
Two Parameters

Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at x=0 are also considered.

Solution of Generalized Jaynes-Cummings Model for many Particle Systems

2013 ◽  
Vol 481 ◽  
pp. 272-277
Author(s):  
N.N. Jr. Bogolubov ◽  
M.Yu. Rasulova ◽  
I.A. Tishabaev
Keyword(s):  
Kinetic Equation ◽  
Linear Equation ◽  
Schrodinger Equation ◽  
Particle Systems ◽  
Cavity Field ◽  
One Dimensional ◽  
Generalized Kinetic Equation ◽  
The One ◽  
The Given ◽  
Multi Mode

We consider the dynamics of a system consisting of N two-level atoms interacting with a multi-mode cavity field. For the given system, the generalized kinetic equation is obtained and conditions are given under which its solution is reduced to solution of a linear equation, and of the one-dimensional nonlinear Schrodinger equation, respectively.

scholarly journals EXACT SOLUTIONS TO THE SPIN-2 GROSS–PITAEVSKII EQUATIONS

2012 ◽  
Vol 27 (02) ◽  
pp. 1350013 ◽  
Author(s):  
ZHI-HAI ZHANG ◽  
YONG-KAI LIU ◽  
SHI-JIE YANG
Keyword(s):  
Exact Solutions ◽  
Rotational Symmetry ◽  
Time Factors ◽  
Time Dependent ◽  
Spin Space ◽  
One Dimensional ◽  
Density Distributions ◽  
Hyperfine State ◽  
The One ◽  
Bose Einstein

We present several exact solutions to the coupled nonlinear Gross–Pitaevskii equations which describe the motion of the one-dimensional spin-2 Bose–Einstein condensates. The nonlinear density–density interactions are decoupled by making use of the properties of Jacobian elliptical functions. The distinct time factors in each hyperfine state implies a "Lamor" procession in these solutions. Furthermore, exact time-evolving solutions to the time-dependent Gross–Pitaevskii equations are constructed through the spin-rotational symmetry of the Hamiltonian. The spin-polarizations and density distributions in the spin-space are analyzed.

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