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Physical Review A

New method for the solution of the logarithmic nonlinear Schrödinger equation via stochastic mechanics

Antonio Boulhosa Nassar

Published May 1, 1986• 30 citations
Publisher

Abstract

Via Nelson's stochastic mechanics, a method for the solution of the hydrodynamical version of the logarithmic nonlinear Schr\"odinger equation (LNLSE), with a time-dependent forced-harmonic-oscillator potential, is presented. Based on a new interpretation of the interplay between dispersion and nonlinearity, a revealing general spreading-wave-packet solution is found. The complete stochastic process associated with the LNLSE is also derived and decomposed into underlying processes of independent nature (classical and quantum). Physical consequences and conditions which allow the existence of solitonlike (nonspreading) solutions are described: Among them, it is found that the zero-point energy is given by ${\ensuremath{\epsilon}}_{0}$=\ensuremath{\Elzxh}${\ensuremath{\Omega}}_{0}$/2, where ${\ensuremath{\Omega}}_{0}$=\ensuremath{\lambda}/2+[(\ensuremath{\lambda}/2)+${\ensuremath{\omega}}_{0}^{2}$] $^{1/2}$ with ${\ensuremath{\omega}}_{0}$ and \ensuremath{\lambda} being the harmonic frequency and the nonlinearity strength, respectively.

DOI
10.1103/physreva.33.3502

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Journal of Mathematical Physics•Oct 1, 1984•50 citations
Via Nelson's stochastic mechanics, a method for the solution of the hydrodynamical version of the logarithmic nonlinear Schr"odinger equation (LNLSE), with a time-dependent forced-harmonic-oscillator potential, is presented. Based on a new interpretation of the interplay between dispersion and nonlinearity, a revealing general spreading-wave-packet solution is found. The complete stochastic process associated with the LNLSE is also derived and decomposed into underlying processes of independent nature (classical and quantum). Physical consequences and conditions which allow the existence of solitonlike (nonspreading) solutions are described: Among them, it is found that the zero-point energy is given by \ensuremathϵ0{\ensuremath{\epsilon}}_{0}\ensuremathϵ0​=\ensuremath{\Elzxh}\ensuremathΩ0{\ensuremath{\Omega}}_{0}\ensuremathΩ0​/2, where \ensuremathΩ0{\ensuremath{\Omega}}_{0}\ensuremathΩ0​=\ensuremath{\lambda}/2+[(\ensuremath{\lambda}/2)+\ensuremathω02{\ensuremath{\omega}}_{0}^{2}\ensuremathω02​] 1/2^{1/2}1/2 with \ensuremathω0{\ensuremath{\omega}}_{0}\ensuremathω0​ and \ensuremath{\lambda} being the harmonic frequency and the nonlinearity strength, respectively.
We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.
Taking into account wave-particle duality and the uncertainty principle, a nonlinear Schrödinger wave equation (NLSE) for the description of dissipative systems can be constructed by applying only Newton's form of classical mechanics. A comparison between our NLSE for dissipative systems and the linear time-dependent SE for conservative systems shows that, for free motion and motion in a harmonic potential, both equations have Gaussian wave-packet solutions. The time dependence of the maximum of these functions is governed by the classical (real) Newtonian equation of motion. While the time dependence of the width of the wave packet can be determined from a nonlinear complex Bernoulli/Ricatti type differential equation. This can be done by transformation into a Newton-type equation for a real quantity α (which is proportional to the square root of the width), including a kind of “centrifugal force” term. This equation can be solved analytically and when taken with Newton's equation of motion for the mean value of the position, it forms a so-called Ermakov system. For the linear SE equation, this implies the existence of a dynamical invariant. For our NLSE we also obtain such an exact invariant, but now for a dissipative system. Furthermore, this complex nonlinear equation can be linearised using the logarithmic derivative of a complex quantity λ, thus yielding a complex Newtonian equation of motion. The absolute value of λ is equivalent to α, the phase angle can be determined using a relation that connects real and imaginary parts of λ. This relation is linked to a kind of “conservation of angular momentum” for the quantity λ in a complex plane. The relations between the above-mentioned quantities will be discussed in detail.
Numerical solution of nonlinear Schrodinger equation (NLSE) can be used to describe an inhomogeneous,weakly interact Bose-Einstein condensate in a small harmonic trap potential at zero temperature. In one dimension situation, we find solutions for the NLSE for ground-state condensate wave function, and obtain the ground-state condensate energy per particle. Furthermore, with the time-dependent NLSE, we can examine the time evolution of the macroscopic wave function even when the trap potential is changed on a time scale comparable to that of the condensate dynamics, a situation that can be easily achieved in magneto-optical traps. We show that there are stable solutions for atomic species with both positive and negative s-wave scattering length in one-dimensional (1D) system for a fixed number of atoms, and the negative scattering length solutions have solitonlike properties.
A new theory for the description of dissipative systems by nonlinear Schrödinger-type field equations (NLSE’s) with logarithmic nonlinearity, which has been recently developed by the authors, is applied to investigate the frictionally damped free motion and similar spatially unrestricted aperiodic problems. Wave-packet solutions as well as time-dependent wave-function solutions are derived and discussed. In the limit of vanishing friction (friction constant γ→0) these solutions turn into the well-known solutions of the respective linear Schrödinger field equation. The same applies to the mean values of position, momentum, and energy, as well as to the uncertainty product of position and momentum. For γ≠0, however, interesting new effects appear. In contrast to the linear theory the uncertainty product of position and momentum does not diverge any more for infinitely long times, t→∞, but asymptotically approaches a definite constant value which depends on characteristic parameters of the system like its mass, initial width, and friction constant γ. Another effect, the faster spreading of the Gaussian wave-packet solution compared to the linear theory, can be explained with the help of a special property of our nonlinear differential equation. In a way similar to what is usually only known for linear differential equations, the wave-packet solutions of our NLSE can be obtained by superposition of the wave functions which are individually also solutions of the same NLSE. The properties of the time-dependent superposition coefficients appearing in this connection are discussed. The extension to the corresponding three-dimensional problem as well as the differences arising in the investigation of the NLSE’s of the free fall and the motion in a constant electric field are given. Concluding, some differences are discussed which appear applying our nonlinear field theory to describe periodic or aperiodic motions, respectively.