Nonlinear Schroedinger equation in the presence of uniform acceleration

Abstract

We consider a recently proposed nonlinear Schroedinger equation exhibiting soliton-like solutions of the power-law form \documentclass[12pt]{minimal}\begin{document}$e_q^{i(kx-wt)}$\end{document}eqi(kx−wt), involving the q-exponential function which naturally emerges within nonextensive thermostatistics [\documentclass[12pt]{minimal}\begin{document}$e_q^z \equiv [1+(1\break-q)z]^{1/(1-q)}$\end{document}eqz≡[1+(1−q)z]1/(1−q), with \documentclass[12pt]{minimal}\begin{document}$e_1^z=e^z$\end{document}e1z=ez]. Since these basic solutions behave like free particles, obeying p = ℏk, E = ℏω, and E = p2/2m (1 ⩽ q < 2), it is relevant to investigate how they change under the effect of uniform acceleration, thus providing the first steps towards the application of the aforementioned nonlinear equation to the study of physical scenarios beyond free particle dynamics. We investigate first the behaviour of the power-law solutions under Galilean transformation and discuss the ensuing Doppler-like effects. We consider then constant acceleration, obtaining new solutions that can be equivalently regarded as describing a free particle viewed from an uniformly accelerated reference frame (with acceleration a) or a particle moving under a constant force −ma. The latter interpretation naturally leads to the evolution equation \documentclass[12pt]{minimal}\begin{document}$i\hbar \frac{\partial }{\partial t}\left( \frac{\Phi }{\Phi _0} \right) \,\, = \,\, - \frac{1}{2-q}\frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} \left[\left( \frac{\Phi }{\Phi _0} \right)^{2-q}\right] + V(x)\left( \frac{\Phi }{\Phi _0} \right)^{q}$\end{document}iℏ∂∂tΦΦ0=−12−qℏ22m∂2∂x2ΦΦ02−q+V(x)ΦΦ0q with V(x) = max. Remarkably enough, the potential V couples to Φq, instead of coupling to Φ, as happens in the familiar linear case (q = 1).