{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T22:16:25Z","timestamp":1770329785732,"version":"3.49.0"},"reference-count":56,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T00:00:00Z","timestamp":1716163200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CONACYT","award":["CVU 1184448"],"award-info":[{"award-number":["CVU 1184448"]}]},{"name":"CONACYT","award":["1695\/22"],"award-info":[{"award-number":["1695\/22"]}]},{"name":"Israel Science Foundation","award":["CVU 1184448"],"award-info":[{"award-number":["CVU 1184448"]}]},{"name":"Israel Science Foundation","award":["1695\/22"],"award-info":[{"award-number":["1695\/22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The propagation of light beams in photovoltaic pyroelectric photorefractive crystals is modelled by a specific generalization of the nonlinear Schr\u00f6dinger equation (GNLSE). We use a variational approximation (VA) to predict the propagation of solitary-wave inputs in the crystals, finding that the VA equations involve a dilogarithm special function. The VA predicts that solitons and breathers exist, and the Vakhitov\u2013Kolokolov criterion predicts that the solitons are stable solutions. Direct simulations of the underlying GNLSE corroborates the existence of such stable modes. The numerical solutions produce both regular breathers and ones featuring beats (long-period modulations of fast oscillations). In the latter case, the Fourier transform of amplitude oscillations reveals a nearly discrete spectrum characterizing the beats dynamics. Numerical solutions of another type demonstrate the spontaneous splitting of the input pulse in two or several secondary ones.<\/jats:p>","DOI":"10.3390\/axioms13050338","type":"journal-article","created":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T11:06:41Z","timestamp":1716203201000},"page":"338","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Regular, Beating and Dilogarithmic Breathers in Biased Photorefractive Crystals"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1865-346X","authenticated-orcid":false,"given":"Carlos Alberto","family":"Betancur-Silvera","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica, Departamento de Sistemas Complejos, Universidad Nacional Aut\u00f3noma de M\u00e9xico, P.O. Box 20-364, Mexico City 01000, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aurea","family":"Espinosa-Cer\u00f3n","sequence":"additional","affiliation":[{"name":"Facultad de Ciencias, Universidad Nacional Aut\u00f3noma de M\u00e9xico, Mexico City 04510, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5323-1847","authenticated-orcid":false,"given":"Boris A.","family":"Malomed","sequence":"additional","affiliation":[{"name":"Department of Physical Electronics, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel"},{"name":"Instituto de Alta Investigaci\u00f3n, Universidad de Tarapac\u00e1, Casilla 7D, Arica 1000000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jorge","family":"Fujioka","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Departamento de Sistemas Complejos, Universidad Nacional Aut\u00f3noma de M\u00e9xico, P.O. Box 20-364, Mexico City 01000, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1063\/1.1754607","article-title":"Optically-induced refractive index inhomogeneities in LiNbO3 and LiTaO3","volume":"9","author":"Ashkin","year":"1966","journal-title":"Appl. Phys. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3211","DOI":"10.1103\/PhysRevLett.73.3211","article-title":"Steady-state spatial screening solitons in photorefractive materials with external applied field","volume":"73","author":"Seguev","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1772","DOI":"10.1364\/JOSAB.14.001772","article-title":"Photovoltaic spatial solitons","volume":"14","author":"Segev","year":"1997","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1364\/JOSAB.16.000550","article-title":"Screening-photovoltaic spatial solitons in biased photovoltaic\u2013photorefractive crystals and their self-deflection","volume":"16","author":"Liu","year":"1999","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3085","DOI":"10.1016\/j.physleta.2012.08.043","article-title":"Pyroelectric photovoltaic spatial solitons in unbiased photorefractive crystals","volume":"376","author":"Jiang","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.physleta.2016.10.054","article-title":"Spatial solitons in biased photovoltaic photorefractive materials with the pyroelectric effect","volume":"381","author":"Katti","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"R4457","DOI":"10.1103\/PhysRevA.50.R4457","article-title":"Dark and bright photovoltaic spatial solitons","volume":"50","author":"Valley","year":"1994","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3095","DOI":"10.1103\/PhysRevA.52.3095","article-title":"Observation of dark photovoltaic spatial solitons","volume":"52","author":"Taya","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2121","DOI":"10.1364\/JOSAB.23.002121","article-title":"Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam","volume":"23","author":"She","year":"2006","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2193","DOI":"10.1063\/1.1794854","article-title":"Screening-photovoltaic bright solitons in lithium niobate and associated single-mode waveguides","volume":"85","author":"Fazio","year":"2004","journal-title":"App. Phys. Lett."},{"key":"ref_11","first-page":"021301","article-title":"Bright optical spatial solitons in a photovoltaic photorefractive waveguide exhibiting the two photon photorefractive effect","volume":"69","author":"Katti","year":"2023","journal-title":"Rev. Mex. F\u00edsica"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"033610","DOI":"10.1103\/PhysRevA.66.033610","article-title":"Localized breathing oscillations of Bose-Einstein condensates in periodic traps","volume":"66","author":"Promislow","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"224101","DOI":"10.1103\/PhysRevLett.102.224101","article-title":"Discrete breathers in a forced-damped array of coupled pendula: Modeling, computation, and experiment","volume":"102","author":"Cuevas","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2855","DOI":"10.1007\/s11071-018-4726-8","article-title":"Breathers, solitons and rogue waves of the quintic nonlinear Schr\u00f6dinger equation on various backgrounds","volume":"95","author":"Ashour","year":"2019","journal-title":"Nonlinear Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2008.05.002","article-title":"Discrete breathers\u2014Advances in theory and applications","volume":"467","author":"Flach","year":"2008","journal-title":"Phys. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"106282","DOI":"10.1016\/j.cnsns.2022.106282","article-title":"Solitons in PT-symmetric systems with spin\u2013orbit coupling and critical nonlinearity","volume":"109","author":"Burlak","year":"2022","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1007\/s12043-015-1093-6","article-title":"Some aspects of optical spatial solitons in photorefractive media and their important applications","volume":"85","author":"Konar","year":"2015","journal-title":"Pramana"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Katti, A., and Yadav, R.A. (2021). Optical Spatial Solitons in Photorefractive Materiales, Springer.","DOI":"10.1007\/978-981-16-2550-3"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Segur, H. (1981). Solitons and the Inverse Scattering Transform, SIAM.","DOI":"10.1137\/1.9781611970883"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Clarkson, P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press.","DOI":"10.1017\/CBO9780511623998"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3135","DOI":"10.1103\/PhysRevA.27.3135","article-title":"Variational approach to nonlinear pulse propagation in optical fibers","volume":"27","author":"Anderson","year":"1983","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1364\/JOSAB.5.000207","article-title":"Asymptotic propagation properties of pulses in a soliton-based optical-fiber communication system","volume":"5","author":"Anderson","year":"1988","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1103\/PhysRevE.48.1418","article-title":"Resonant shape oscillations and decay of a soliton in a periodically inhomogeneous nonlinear optical fiber","volume":"48","author":"Malomed","year":"1993","journal-title":"Phys. Rev. E"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1103\/PhysRevE.51.1484","article-title":"Soliton evolution and radiation loss for the nonlinear Schr\u00f6dinger equation","volume":"51","author":"Kath","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1103\/PhysRevE.51.709","article-title":"Soliton interactions on dual-core fibers","volume":"51","author":"Doty","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6287","DOI":"10.1103\/PhysRevA.41.6287","article-title":"Approximate model of soliton dynamics in all-optical couplers","volume":"41","author":"Florjanczyk","year":"1990","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2039","DOI":"10.1016\/j.physleta.2019.03.043","article-title":"Variational approximations of soliton dynamics in the Ablowitz-Musslimani nonlinear Schr\u00f6dinger equation","volume":"383","author":"Rusin","year":"2019","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2440","DOI":"10.1143\/JPSJ.65.2440","article-title":"Stability of the bright-type algebraic solitary-wave solutions of two extended versions of the nonlinear Schr\u00f6dinger equation","volume":"65","author":"Fujioka","year":"1996","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2601","DOI":"10.1143\/JPSJ.66.2601","article-title":"Soliton-like solution of an extended NLS equation existing in resonance with linear dispersive waves","volume":"66","author":"Fujioka","year":"1997","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"034007","DOI":"10.7566\/JPSJ.82.034007","article-title":"Radiationless higher-order embedded solitons","volume":"82","author":"Fujioka","year":"2013","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"111798","DOI":"10.1016\/j.chaos.2022.111798","article-title":"Lagrangian nonlocal nonlinear Schr\u00f6dinger equations","volume":"156","author":"Fujioka","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/S0079-6638(02)80026-9","article-title":"Variational methods in nonlinear fiber optics and related fields","volume":"43","author":"Malomed","year":"2002","journal-title":"Progr. Optics"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Malomed, B.A. (2022). Multidimensional Solitons, American Institute of Physics Publishing.","DOI":"10.1063\/9780735425118"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gunter, P., and Huignard, J.P. (1988). Photorefractive Materials and Their Applications I and II, Springer.","DOI":"10.1007\/3-540-18332-9_29"},{"key":"ref_35","unstructured":"Yeh, P. (1993). Photorefractive Nonlinear Optics, Wiley."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.1016\/j.ijleo.2013.07.152","article-title":"Coherent interactions of multi bright spatial solitons in biased photorefractive crystals","volume":"125","author":"Su","year":"2014","journal-title":"Optik"},{"key":"ref_37","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, Academic Press."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"063822","DOI":"10.1103\/PhysRevA.98.063822","article-title":"Rotating solitons supported by a spiral waveguide","volume":"98","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"25757","DOI":"10.1364\/OE.18.025757","article-title":"Energy scalability of mode-locked oscillators: A completely analytical approach to analysis","volume":"18","author":"Kalashnikov","year":"2010","journal-title":"Opt. Express"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/BF01031343","article-title":"Stationary solutions of the wave equation in a medium with nonlinearity saturation","volume":"16","author":"Vakhitov","year":"1973","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.1007\/BF01036894","article-title":"Stability of stationary solutions of nonlinear wave equations","volume":"17","author":"Kolokolov","year":"1974","journal-title":"Radiophys. Quantum Electron."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Yang, J. (2010). Nonlinear Waves in Integrable and Nonintegrable Systems, SIAM.","DOI":"10.1137\/1.9780898719680"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"033834","DOI":"10.1103\/PhysRevA.81.033834","article-title":"Band-gap boundaries and fundamental solitons in complex two-dimensional nonlinear lattices","volume":"81","author":"Ablowitz","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"043604","DOI":"10.1103\/PhysRevA.63.043604","article-title":"Stability of Trpped Bose-Einstein condensates","volume":"63","author":"Abdullaev","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0167-2789(95)00149-X","article-title":"Nonlinear interaction of solitons and radiation","volume":"87","author":"Kuznetsov","year":"1995","journal-title":"Physica D"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1143\/PTPS.55.284","article-title":"Initial Value problems of one-dimensional self-modulation of nonlinear waves in dispersive media","volume":"55","author":"Satsuma","year":"1974","journal-title":"Suppl. Prog. Theor. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"016605","DOI":"10.1103\/PhysRevE.66.016605","article-title":"Solitonlike propagation in photorefractive crystals with large optical activity and absorption","volume":"66","author":"Fazio","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"205303","DOI":"10.1088\/1361-6455\/aae0ba","article-title":"Probing quasi-integrability of the Gross-Pitaevskii equation in a harmonic-oscillator potential","volume":"51","author":"Bland","year":"2018","journal-title":"J. Phys. B"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"4293","DOI":"10.1007\/s11071-021-06534-9","article-title":"Analytical solutions of solitary waves and their collision stability in a pre-compressed one-dimensional granular crystal","volume":"104","author":"Liu","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"244302","DOI":"10.1103\/PhysRevLett.104.244302","article-title":"Discrete breathers in one-dimensional diatomic granular crystals","volume":"104","author":"Boechler","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"413003","DOI":"10.1088\/1361-648X\/aa7672","article-title":"Nonlinear coherent structures in granular crystals","volume":"29","author":"Chong","year":"2017","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"446","DOI":"10.3367\/UFNe.2016.02.037729","article-title":"Discrete breathers in crystals","volume":"59","author":"Dimitriev","year":"2016","journal-title":"Phys.-Uspekhi"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"014220","DOI":"10.1103\/PhysRevE.107.014220","article-title":"Discrete breathers in a mechanical metamaterial","volume":"107","author":"Duran","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1103\/PhysRevLett.83.248","article-title":"Discrete Breathers in Systems with Homogeneous Potentials: Analytic Solutions","volume":"83","author":"Ovchinnikov","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1103\/PhysRevE.50.1635","article-title":"Effect of chirp on soliton production","volume":"50","author":"Kaup","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"253901","DOI":"10.1103\/PhysRevLett.107.253901","article-title":"Higher-order modulation instability in nonlinear fiber optics","volume":"107","author":"Erkintalo","year":"2011","journal-title":"Phys. Rev. Lett."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/338\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:45:23Z","timestamp":1760107523000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/338"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,20]]},"references-count":56,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["axioms13050338"],"URL":"https:\/\/doi.org\/10.3390\/axioms13050338","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,20]]}}}