{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T05:40:41Z","timestamp":1768801241561,"version":"3.49.0"},"reference-count":103,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2017,8,18]],"date-time":"2017-08-18T00:00:00Z","timestamp":1503014400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"GA\\v{C}R","award":["16-22945S"],"award-info":[{"award-number":["16-22945S"]}]},{"name":"National Research Foundation of South Africa","award":["95965, 98083 and 98892"],"award-info":[{"award-number":["95965, 98083 and 98892"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Schr\u00f6dinger equations with non-Hermitian, but    PT   -symmetric quantum potentials     V ( x )     found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an \u201canomalous\u201d refraction index. This turned attention to the nonlinear Schr\u00f6dinger equations in which the interaction term becomes state-dependent,     V ( x ) \u2192 W ( \u03c8 ( x ) , x )    . Here, the state-dependence in     W ( \u03c8 ( x ) , x )     is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction     W ( \u03c8 ( x ) , x )     is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by    PT   -symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schr\u00f6dinger equations.<\/jats:p>","DOI":"10.3390\/sym9080165","type":"journal-article","created":{"date-parts":[[2017,8,21]],"date-time":"2017-08-21T04:12:17Z","timestamp":1503288737000},"page":"165","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Schr\u00f6dinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels"],"prefix":"10.3390","volume":"9","author":[{"given":"Miloslav","family":"Znojil","sequence":"first","affiliation":[{"name":"Nuclear Physics Institute of the CAS, Hlavn\u00ed 130, 250 68 \u0158e\u017e, Czech Republic"},{"name":"Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Franti\u0161ek","family":"R\u016f\u017ei\u010dka","sequence":"additional","affiliation":[{"name":"Nuclear Physics Institute of the CAS, Hlavn\u00ed 130, 250 68 \u0158e\u017e, Czech Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9960-2874","authenticated-orcid":false,"given":"Konstantin","family":"Zloshchastiev","sequence":"additional","affiliation":[{"name":"Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,8,18]]},"reference":[{"key":"ref_1","unstructured":"Messiah, A. 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