Nonlinear q-Generalizations of Quantum Equations: Homogeneous and Nonhomogeneous Cases—An Overview
Recent developments on the generalizations of two important equations of quantum physics, namely the Schroedinger and Klein-Gordon equations, are reviewed. These generalizations present nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard linear equations are recovered in the limit q → 1. Interestingly, these equations present a common, soliton-like, traveling solution, which is written in terms of the q-exponential function that naturally emerges
... within nonextensive statistical mechanics. In both cases, the corresponding well-known Einstein energy-momentum relations, as well as the Planck and the de Broglie ones, are preserved for arbitrary values of q. In order to deal appropriately with the continuity equation, a classical field theory has been developed, where besides the usual Ψ( x, t), a new field Φ( x, t) must be introduced; this latter field becomes Ψ * ( x, t) only when q → 1. A class of linear nonhomogeneous Schroedinger equations, characterized by position-dependent masses, for which the extra field Φ( x, t) becomes necessary, is also investigated. In this case, an appropriate transformation connecting Ψ( x, t) and Φ( x, t) is proposed, opening the possibility for finding a connection between these fields in the nonlinear cases. The solutions presented herein are potential candidates for applications to nonlinear excitations in plasma physics, nonlinear optics, in structures, such as those of graphene, as well as in shallow and deep water waves. Recently, NL equations have attracted much interest in science due to their potential for describing a wide variety of phenomena, more specifically those within the realm of complex systems [68, 69] . In fact, the applicability of linear equations is usually restricted to idealized systems, being valid for media characterized by specific conditions, like homogeneity, isotropy and translational invariance, with particles interacting through short-range forces and with a dynamical behavior characterized by Entropy 2017, 19, 39 3 of 25 short-time memory. However, many real systems do not fulfil these requirements and usually exhibit complicated collective behavior associated with NL phenomena. Since finding analytical solutions of NL equations may become a hard task, particularly in the case of NL differential equations , very frequently, one has to make use of numerical procedures, and so, a considerable advance has been attained lately in this area. Among the most studied NL differential equations, we mention the Fokker-Planck , as well as two distinct proposals for the Schroedinger one. In the Fokker-Planck case, the nonlinearity was firstly introduced through a power in the probability of the diffusion term, in such a way as to modify it into a nonlinear diffusion term [72, 73] . Consequently, in the same way that the linear Fokker-Planck equation is associated to normal diffusion and to the BG entropy, the NL Fokker-Planck proposal of [72, 73] is connected to anomalous-diffusion phenomena and to the nonadditive entropy S q of Equation (3) , which yields nonextensive statistical mechanics [6, 9, 10] . Indeed, by means of generalized forms of the H-theorem, it is possible to connect very general types of NL Fokker-Planck equations to entropic forms    . As concerns the Schroedinger equation, there are essentially two most investigated NL proposals in the literature: (i) a previous one, corresponding to the introduction of an extra term containing a power in the wave function (usually a cubic one)    77 ]; (ii) in a more recent proposal, the same procedure used for the NL Fokker-Planck of [72, 73] was considered, namely by introducing a power in the wave function of an existing term  . In both cases, one has compact traveling solutions, characterized by a spatial part that does not deform throughout the evolution. An important property of these types of solutions concerns their square integrability, allowing for an appropriate normalization. Due to the modulation of the wave function, these solutions are considered to be relevant in diverse areas of physics, including nonlinear optics, superconductivity, plasma physics and deep water waves [68, 69] . Herein, we will restrict ourselves to this second proposal, analyzing the similarities and differences with respect to the linear case and reviewing the most recent studies related to this equation. Moreover, we will also discuss briefly proposals for linear nonhomogeneous Schroedinger equations, as well as an NL generalization of the Klein-Gordon equation introduced in . Next, we highlight some basic results of the linear Schroedinger equation, which will be useful for the discussion of the nonlinear case.