# Perturbative Treatment of the Non-Linear q-Schrödinger and q-Klein–Gordon Equations

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La Plata National University and Argentina’s National Research Council, (IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata, Argentina

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Faculty of Exact and Natural Sciences, La Pampa National University, Uruguay 151, Santa Rosa, 3300 La Pampa, Argentina

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Author to whom correspondence should be addressed.

Academic Editor: Kevin H. Knuth

Received: 5 November 2016 / Revised: 28 December 2016 / Accepted: 29 December 2016 / Published: 31 December 2016

(This article belongs to the Special Issue Entropic Aspects of Nonlinear Partial Differential Equations: Classical and Quantum Mechanical Perspectives)

Interesting non-linear generalization of both Schrödinger’s and Klein–Gordon’s equations have been recently advanced by Tsallis, Rego-Monteiro and Tsallis (NRT) in Nobre et al. (Phys. Rev. Lett. 2011, 106, 140601). There is much current activity going on in this area. The non-linearity is governed by a real parameter q. Empiric hints suggest that the ensuing non-linear q-Schrödinger and q-Klein–Gordon equations are a natural manifestations of very high energy phenomena, as verified by LHC-experiments. This happens for $q-$ values close to unity (Plastino et al. (Nucl. Phys. A 2016, 955, 16–26, Nucl. Phys. A 2016, 948, 19–27)). It might thus be difficult for q-values close to unity to ascertain whether one is dealing with solutions to the ordinary Schrödinger equation (whose free particle solutions are exponentials and for which $q=1$ ) or with its NRT non-linear 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.