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

Via Nelson’s stochastic mechanics, a method for the solution of the hydrodynamical version of the logarithmic nonlinear Schrödinger 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 ${\epsilon }_0$=ħ${\Omega }_0$/2, where ${\Omega }_0$=λ/2+[(λ/2)+${\omega }_0^2$] ${}_{}{}^{1/2}{}_{}{}^{}$ with ${\omega }_0$ and λ being the harmonic frequency and the nonlinearity strength, respectively.