Completely Integrable Equation for the Quantum Correlation Function of Nonlinear Schrödinger Equation

SynopsisIn this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equationwhere a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only ifwhere α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

Download Full-text

DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS

International Journal of Modern Physics B ◽

10.1142/s0217979290000504 ◽

1990 ◽

Vol 04 (05) ◽

pp. 1003-1037 ◽

Cited By ~ 179

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.

Download Full-text

O(N)-invariant nonlinear Schrödinger equation — A new completely integrable system

Physics Letters A ◽

10.1016/0375-9601(81)90205-x ◽

1981 ◽

Vol 84 (7) ◽

pp. 349-352 ◽

Cited By ~ 29

Author(s):

P.P. Kulish ◽

E.K. Sklyanin

Keyword(s):

Schrödinger Equation ◽

Integrable System ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Nonlinear Schrödinger ◽

Completely Integrable ◽

Completely Integrable System

Download Full-text

SYMPLECTIC NUMERICAL METHODS IN DYNAMICS OF NONLINEAR WAVES

International Journal of Modern Physics C ◽

10.1142/s0129183199000760 ◽

1999 ◽

Vol 10 (05) ◽

pp. 967-980 ◽

Cited By ~ 3

Author(s):

A. G. SHAGALOV

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Integrable Equation ◽

Nonlinear Schrodinger Equation ◽

Symplectic Integrators ◽

Nonlinear Schrödinger ◽

Higher Derivative ◽

Long Time ◽

Derivative Nonlinear Schrödinger Equation

The symplectic integrator of the Gauss–Legendre type is tested on the nonlinear Schrödinger equation. Preservation of high integrals (up to 10 or more) and quasiperiodic motion have been detected for dynamics on both stable soliton and homoclinic manifolds, which indicate applicability of symplectic integrators for adequate simulation of integrable equation. The tested integrator is applied to the problem of long-time stability of the solitons in higher-derivative nonlinear Schrödinger equation. The slow logarithmic-type depletion of the soliton amplitude with time has been detected.

Download Full-text