A Family of Exact, Analytic Time Dependent Wave Packet Solutions to a Nonlinear Schroedinger Equation

TL;DR

This paper derives exact, time-dependent q-Gaussian wave packet solutions for a specific nonlinear Schrödinger equation inspired by nonextensive thermostatistics, extending known solutions and highlighting their unique features compared to other nonlinear models.

Contribution

It provides explicit analytical solutions for the NRT nonlinear Schrödinger equation, demonstrating their relation to known Gaussian solutions and establishing their uniqueness among similar nonlinear equations.

Findings

Derived exact q-Gaussian wave packet solutions

Extended the class of solutions beyond q-plane waves

Showed the uniqueness of Gaussian-like solutions for the NRT equation

Abstract

We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schr\"odinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane wave-like solutions ($q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the $q$-generalized thermostatistical formalism, is characterized by a parameter $q$, and in the limit $q \to 1$ reduces to the standard, linear Schr\"odinger equation. The $q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known $q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the $q \to 1$ limit the $q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schr\"odinger equation. In the present work we also show that there are other families of nonlinear Schr\"odinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.