Perturbative treatment of the non-linear q-Schr\"odinger and q-Klein-Gordon equations

Interesting nonlinear generalization of both Schr\"odinger's and Klein-Gordon's equations have been recently advanced by Tsallis, Rego-Monteiro, and Tsallis (NRT) in [Phys. Rev. Lett. {\bf 106}, 140601 (2011)]. There is much current activity going on in this area. The non-linearity is governed by a real parameter $q$. It is a fact that the ensuing non linear q-Schr\"odinger and q-Klein-Gordon equations are natural manifestations of very high energy phenomena, as verified by LHC-experiments. This happens for $q-$values close to unity [Nucl. Phys. A {\bf 955}, 16 (2016), Nucl. Phys. A {\bf 948}, 19 (2016)]. It is also well known that q-exponential behavior is found in quite different settings. An explanation for such phenomenon was given in [Physica A {\bf 388}, 601 (2009)] with reference to empirical scenarios in which data are collected via set-ups that effect a normalization plus data's pre-processing. Precisely, the ensuing normalized output was there shown to be q-exponentially distributed if the input data display elliptical symmetry, generalization of spherical symmetry, a frequent situation. This makes it difficult, for q-values close to unity, to ascertain whether one is dealing with solutions to the ordinary Schr\"odinger equation (whose free particle solutions are exponentials, and for which $q=1$) or with its NRT nonlinear q-generalizations, whose free particle solutions are q-exponentials. In this work we provide a careful analysis of the $q \sim 1$ instance via a perturbative analysis of the NRT equations.