{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:20:45Z","timestamp":1760242845269,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,12,31]],"date-time":"2016-12-31T00:00:00Z","timestamp":1483142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002923","name":"Consejo Nacional de Investigaciones Cient\u00edficas y T\u00e9cnicas","doi-asserted-by":"publisher","award":["PIP 029\/12"],"award-info":[{"award-number":["PIP 029\/12"]}],"id":[{"id":"10.13039\/501100002923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Interesting non-linear generalization of both Schr\u00f6dinger\u2019s and Klein\u2013Gordon\u2019s 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-Schr\u00f6dinger and q-Klein\u2013Gordon equations are a natural manifestations of very high energy phenomena, as verified by LHC-experiments. This happens for     q \u2212    values close to unity (Plastino et al. (Nucl. Phys. A 2016, 955, 16\u201326, Nucl. Phys. A 2016, 948, 19\u201327)). It might thus be difficult for q-values close to unity to ascertain whether one is dealing with solutions to the ordinary Schr\u00f6dinger 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 \u223c 1     instance via a perturbative analysis of the NRT equations.<\/jats:p>","DOI":"10.3390\/e19010021","type":"journal-article","created":{"date-parts":[[2017,1,2]],"date-time":"2017-01-02T09:54:18Z","timestamp":1483350858000},"page":"21","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Perturbative Treatment of the Non-Linear q-Schr\u00f6dinger and q-Klein\u2013Gordon Equations"],"prefix":"10.3390","volume":"19","author":[{"given":"Javier","family":"Zamora","sequence":"first","affiliation":[{"name":"La Plata National University and Argentina\u2019s National Research Council, (IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mario","family":"Rocca","sequence":"additional","affiliation":[{"name":"La Plata National University and Argentina\u2019s National Research Council, (IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Angelo","family":"Plastino","sequence":"additional","affiliation":[{"name":"La Plata National University and Argentina\u2019s National Research Council, (IFLP-CCT-CONICET)-C. C. 727, 1900 La Plata, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gustavo","family":"Ferri","sequence":"additional","affiliation":[{"name":"Faculty of Exact and Natural Sciences, La Pampa National University, Uruguay 151, Santa Rosa, 3300 La Pampa, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,12,31]]},"reference":[{"key":"ref_1","unstructured":"Boccara, N. (2004). Modeling Complex Systems, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1016\/j.physa.2008.11.001","article-title":"Why is the detection of q-Gaussian behavior such a common occurrence?","volume":"388","author":"Vignat","year":"2009","journal-title":"Physica A"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2690","DOI":"10.1016\/j.physleta.2015.08.015","article-title":"From the hypergeometric differential equation to a non-linear Schroedinger one","volume":"379","author":"Plastino","year":"2015","journal-title":"Phys. Lett. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"140601","DOI":"10.1103\/PhysRevLett.106.140601","article-title":"Nonlinear generalizations of relativistic and quantum equations with a common type of solution","volume":"106","author":"Nobre","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"062134","DOI":"10.1103\/PhysRevA.90.062134","article-title":"Stationary and uniformly accelerated states in non-linear quantum mechanics","volume":"90","author":"Plastino","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"041505","DOI":"10.1063\/1.4798999","article-title":"Nonlinear Schroedinger equation in the presence of uniform acceleration","volume":"54","author":"Plastino","year":"2013","journal-title":"J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"082106","DOI":"10.1063\/1.4960723","article-title":"Travelling-wave and separated variable solutions of a nonlinear Schroedinger equation","volume":"57","author":"Bountis","year":"2016","journal-title":"J. Math. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.nuclphysa.2016.01.047","article-title":"q-Gamow States as continuous linear functionals on analytical test functions","volume":"948","author":"Plastino","year":"2016","journal-title":"Nucl. Phys. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.nuclphysa.2016.05.021","article-title":"q-Gamow States for intermediate energies","volume":"955","author":"Plastino","year":"2016","journal-title":"Nuc. Phys. A"},{"key":"ref_10","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/j.physa.2016.01.022","article-title":"Hypergeometric connotations of quantum equations","volume":"450","author":"Plastino","year":"2016","journal-title":"Physica A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/1\/21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:29:38Z","timestamp":1760210978000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/19\/1\/21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,31]]},"references-count":11,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1]]}},"alternative-id":["e19010021"],"URL":"https:\/\/doi.org\/10.3390\/e19010021","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,12,31]]}}}