{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T06:19:11Z","timestamp":1785824351864,"version":"3.56.0"},"reference-count":157,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T00:00:00Z","timestamp":1706832000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003977","name":"Israel Science Foundation","doi-asserted-by":"publisher","award":["1695\/22"],"award-info":[{"award-number":["1695\/22"]}],"id":[{"id":"10.13039\/501100003977","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schr\u00f6dinger (DNLS) equations and their generalizations, such as a system of discrete Gross\u2013Pitaevskii (GP) equations with the Lee\u2013Huang\u2013Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole\u2013dipole and quadrupole\u2013quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (\u03c7(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin\u2013orbit-coupled (SOC) binary Bose\u2013Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.<\/jats:p>","DOI":"10.3390\/e26020137","type":"journal-article","created":{"date-parts":[[2024,2,2]],"date-time":"2024-02-02T09:24:08Z","timestamp":1706865848000},"page":"137","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5323-1847","authenticated-orcid":false,"given":"Boris","family":"Malomed","sequence":"first","affiliation":[{"name":"Instituto de Alta Investigaci\u00f3n, Universidad de Tarapac\u00e1, Casilla 7D, Arica 1000000, Chile"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,2]]},"reference":[{"key":"ref_1","unstructured":"Pitaevskii, L.P., and Stringari, S. (2003). Bose\u2013Einstein Condensation, Oxford University Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kivshar, Y.S., and Agrawal, G.P. (2003). Optical Solitons: From Fibers to Photonic Crystals, Academic Press.","DOI":"10.1016\/B978-012410590-4\/50012-7"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Fibich, G. (2015). The Nonlinear Schr\u00f6dinger Equation: Singular Solutions and Optical Collapse, Springer.","DOI":"10.1007\/978-3-319-12748-4"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1103\/RevModPhys.78.179","article-title":"Dynamics of Bose\u2013Einstein condensates in optical lattice","volume":"78","author":"Morsch","year":"2006","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"015115","DOI":"10.1063\/1.1858114","article-title":"Nonlinear lattice dynamics of Bose\u2013Einstein condensates","volume":"15","author":"Porter","year":"2005","journal-title":"Chaos"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/S0079-6638(08)00004-8","article-title":"Soliton shape and mobility control in optical lattices","volume":"52","author":"Kartashov","year":"2009","journal-title":"Prog. Opt."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Skorobogatiy, M., and Yang, J. (2008). Fundamentals of Photonic Crystal Guiding, Cambridge University Press.","DOI":"10.1017\/CBO9780511575228"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1364\/OL.13.000794","article-title":"Discrete self-focusing in nonlinear arrays of coupled waveguides","volume":"13","author":"Christodoulides","year":"1988","journal-title":"Opt. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3383","DOI":"10.1103\/PhysRevLett.81.3383","article-title":"Discrete spatial optical solitons in waveguide arrays","volume":"81","author":"Eisenberg","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1038\/nature01936","article-title":"Discretizing light behaviour in linear and nonlinear waveguide lattices","volume":"424","author":"Christodoulides","year":"2003","journal-title":"Nature"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2008.04.004","article-title":"Discrete solitons in optics","volume":"463","author":"Lederer","year":"2008","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"106802","DOI":"10.1103\/PhysRevLett.104.106802","article-title":"Subwavelength plasmonic lattice solitons in arrays of metallic nanowires","volume":"104","author":"Ye","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"023613","DOI":"10.1103\/PhysRevA.68.023613","article-title":"Nonlinear tight-binding approximation for Bose\u2013Einstein condensates in a lattice","volume":"68","author":"Smerzi","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"046608","DOI":"10.1103\/PhysRevE.66.046608","article-title":"Wannier functions analysis of the nonlinear Schr\u00f6dinger equation with a periodic potential","volume":"66","author":"Alfimov","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1419","DOI":"10.1103\/RevModPhys.84.1419","article-title":"Maximally localized Wannier functions: Theory and applications","volume":"84","author":"Marzari","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2838","DOI":"10.1364\/OL.34.002838","article-title":"Observation of discrete solitons in lattices with second-order interaction","volume":"34","author":"Szameit","year":"2009","journal-title":"Opt. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1205","DOI":"10.1016\/j.physd.2011.04.011","article-title":"Variational approximations in discrete nonlinear Schr\u00f6dinger equations with next-nearest-neighbor couplings","volume":"240","author":"Chong","year":"2011","journal-title":"Phys. D"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"043804","DOI":"10.1103\/PhysRevA.77.043804","article-title":"Long-range interaction in waveguide lattices","volume":"77","author":"Szameit","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.optcom.2004.03.081","article-title":"All optical switching in ultrashort photonic crystal couplers","volume":"237","author":"Locatelli","year":"2004","journal-title":"Opt. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"066606","DOI":"10.1103\/PhysRevE.76.066606","article-title":"Symmetry breaking in linearly coupled dynamical lattices","volume":"76","author":"Herring","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/S0167-2789(96)00261-8","article-title":"Breathers in nonlinear lattices: Existence, linear stability and quantization","volume":"103","author":"Aubry","year":"1997","journal-title":"Phys. D"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1140\/epjst\/e2016-02648-1","article-title":"Nonlinear wave propagation in discrete and continuous systems","volume":"225","author":"Rothos","year":"2016","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"055803","DOI":"10.1088\/1361-6404\/aacca8","article-title":"Introduction to nonlinear discrete systems: Theory and modelling","volume":"39","author":"Tsoy","year":"2018","journal-title":"Eur. J. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Kevrekidis, P.G., Cuevas-Maraver, J., and Saxena, A. (2020). Nonlinearity and discreteness: Solitons in lattices. Emerging Frontiers in Nonlinear Science, Springer Nature.","DOI":"10.1007\/978-3-030-44992-6"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Kevrekidis, P.G. (2009). The Discrete Nonlinear Schr\u00f6dinger Equation: Mathematical Analysis, Numerical Computations, and Physical Perspectives, Springer.","DOI":"10.1007\/978-3-540-89199-4"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1103\/PhysRevLett.73.1055","article-title":"Stability of discrete solitons and quasicollapse to intrinsically localized modes","volume":"73","author":"Laedke","year":"1994","journal-title":"Phys. Rev. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1038\/s42254-019-0025-7","article-title":"Frontiers in multidimensional self-trapping of nonlinear fields and matter","volume":"1","author":"Kartashov","year":"2019","journal-title":"Nat. Rev. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Malomed, B.A. (2022). Multidimensional Solitons, AIP Publishing.","DOI":"10.1063\/9780735425118"},{"key":"ref_29","first-page":"2301592","article-title":"Multidimensional Soliton Systems","volume":"9","author":"Malomed","year":"2024","journal-title":"Adv. Phys. X"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1135","DOI":"10.1103\/PhysRev.106.1135","article-title":"Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low temperature properties","volume":"106","author":"Lee","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"155302","DOI":"10.1103\/PhysRevLett.115.155302","article-title":"Quantum mechanical stabilization of a collapsing Bose-Bose mixture","volume":"115","author":"Petrov","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1126\/science.aao5686","article-title":"Quantum liquid droplets in a mixture of Bose\u2013Einstein condensates","volume":"359","author":"Cabrera","year":"2018","journal-title":"Science"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"135301","DOI":"10.1103\/PhysRevLett.120.135301","article-title":"Bright soliton to quantum droplet transition in a mixture of Bose\u2013Einstein condensates","volume":"120","author":"Cheiney","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"235301","DOI":"10.1103\/PhysRevLett.120.235301","article-title":"Self-bound quantum droplets of atomic mixtures in free space?","volume":"120","author":"Semeghini","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"100401","DOI":"10.1103\/PhysRevLett.117.100401","article-title":"Ultradilute low-dimensional liquids","volume":"117","author":"Petrov","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"01363","DOI":"10.1103\/PhysRevA.98.013631","article-title":"Dynamics of one-dimensional quantum droplets","volume":"98","author":"Astrakharchik","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zakharov, V.E., Manakov, S.V., Novikov, S.P., and Pitaevskii, L.P. (1980). Solitons: The Inverse Scattering Method, Nauka Publishers. English Translation: Consultants Bureau: New York, NY, USA, 1984.","DOI":"10.1007\/978-3-642-81448-8_7"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., and Segur, H. (1981). Solitons and Inverse Scattering Method, SIAM.","DOI":"10.1137\/1.9781611970883"},{"key":"ref_39","unstructured":"Calogero, F., and Degasperis, A. (1982). Spectral Transform and Solitons: Tools to Solve and Investigate Nonlinear Evolution Equations, North-Holland."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Newell, A.C. (1985). Solitons in Mathematics and Physics, SIAM.","DOI":"10.1137\/1.9781611970227"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1137\/0150021","article-title":"On homoclinic structure and numerically induced chaos for the nonlinear Schr\u00f6dinger equation","volume":"50","author":"Ablowitz","year":"1990","journal-title":"SIAM J. Appl. Math."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"10003","DOI":"10.1209\/0295-5075\/84\/10003","article-title":"On the integrability of the discrete nonlinear Schr\u00f6dinger equation","volume":"84","author":"Levi","year":"2008","journal-title":"Europhys. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.1063\/1.523009","article-title":"Nonlinear differential\u2013difference equations and Fourier analysis","volume":"17","author":"Ablowitz","year":"1976","journal-title":"J. Math. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Suris, Y.B. (2003). The Problem of Integrable Discretization: Hamiltonian Approach, Birkhauser.","DOI":"10.1007\/978-3-0348-8016-9"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0167-2789(93)90020-2","article-title":"Solitons on lattices","volume":"68","author":"Duncan","year":"1993","journal-title":"Phys. D"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1143\/JPSJ.22.431","article-title":"Vibration of a chain with a non-linear interaction","volume":"22","author":"Toda","year":"1967","journal-title":"J. Phys. Soc. Jpn."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/0375-9601(92)90058-T","article-title":"A new method to solve the quantum Ablowitz\u2013Ladik system","volume":"162","author":"Salerno","year":"1992","journal-title":"Phys. Lett. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"066001","DOI":"10.1088\/0034-4885\/78\/6\/066001","article-title":"Non-standard Hubbard models in optical lattices: A review","volume":"78","author":"Dutta","year":"2015","journal-title":"Rep. Prog. Phys."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"4131","DOI":"10.1103\/PhysRevE.53.4131","article-title":"Perturbation theories of a discrete, integrable nonlinear Schr\u00f6dinger equation","volume":"53","author":"Cai","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"036608","DOI":"10.1103\/PhysRevE.73.036608","article-title":"Solitons in the Salerno model with competing nonlinearities","volume":"73","author":"Malomed","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"4776","DOI":"10.1103\/PhysRevLett.77.4776","article-title":"Breather mobility in discrete \u03c64 nonlinear lattices","volume":"77","author":"Chen","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"035602","DOI":"10.1103\/PhysRevE.84.035602","article-title":"Bright solitons from defocusing nonlinearities","volume":"84","author":"Borovkova","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"032905","DOI":"10.1103\/PhysRevE.88.032905","article-title":"Discrete localized modes supported by an inhomogeneous defocusing nonlinearity","volume":"88","author":"Maluckov","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"043201","DOI":"10.1103\/PhysRevE.91.043201","article-title":"Solitons and vortices in two-dimensional discrete nonlinear Schr\u00f6dinger systems with spatially modulated nonlinearity","volume":"91","author":"Kevrekidis","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"200404","DOI":"10.1103\/PhysRevLett.95.200404","article-title":"Two-dimensional bright solitons in dipolar Bose\u2013Einstein condensates","volume":"95","author":"Pedri","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"126401","DOI":"10.1088\/0034-4885\/72\/12\/126401","article-title":"The physics of dipolar bosonic quantum gases","volume":"72","author":"Lahaye","year":"2009","journal-title":"Rep. Prog. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"055303","DOI":"10.1088\/0953-4075\/43\/5\/055303","article-title":"Discrete vortex solitons in dipolar Bose\u2013Einstein condensates","volume":"43","author":"Maluckov","year":"2010","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"090406","DOI":"10.1103\/PhysRevLett.100.090406","article-title":"Anisotropic solitons in dipolar Bose\u2013Einstein condensates","volume":"100","author":"Tikhonenkov","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"063635","DOI":"10.1103\/PhysRevA.88.063635","article-title":"Lattice solitons with quadrupolar intersite interactions","volume":"88","author":"Li","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/S0370-1573(02)00196-5","article-title":"Optical solitons due to quadratic nonlinearities: From basic physics to futuristic applications","volume":"370","author":"Buryak","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"214103","DOI":"10.1103\/PhysRevLett.99.214103","article-title":"Mobility of discrete solitons in quadratically nonlinear media","volume":"99","author":"Susanto","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"3198","DOI":"10.1103\/PhysRevA.46.3198","article-title":"Modulational instabilities in discrete lattices","volume":"46","author":"Kivshar","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_63","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":"Prog. Opt."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/0375-9601(96)00516-6","article-title":"Soliton dynamics in the discrete nonlinear Schr\u00f6dinger equation","volume":"220","author":"Malomed","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"046604","DOI":"10.1103\/PhysRevE.68.046604","article-title":"Soliton collisions in the discrete nonlinear Schr\u00f6dinger equation","volume":"68","author":"Papacharalampous","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1016\/j.matcom.2005.01.015","article-title":"Variational solutions for the discrete nonlinear Schr\u00f6dinger equation","volume":"69","author":"Kaup","year":"2005","journal-title":"Math. Comput. Simulat."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"026604","DOI":"10.1103\/PhysRevE.85.026604","article-title":"Unstaggered-staggered solitons in two-component discrete nonlinear Schr\u00f6dinger lattices","volume":"85","author":"Malomed","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"124","DOI":"10.2991\/jnmp.2008.15.s3.13","article-title":"Approximation of solitons in the discrete NLS equation","volume":"15","author":"Cuevas","year":"2008","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.physd.2011.10.004","article-title":"On the validity of the variational approximation in discrete nonlinear Schr\u00f6dinger equations","volume":"241","author":"Chong","year":"2012","journal-title":"Phys. D"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"5020","DOI":"10.1103\/PhysRevE.50.5020","article-title":"Dark solitons in discrete lattices","volume":"50","author":"Kivshar","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1209\/epl\/i2002-00599-0","article-title":"Matter solitons in Bose\u2013Einstein condensates with optical lattices","volume":"58","author":"Alfimov","year":"2002","journal-title":"Europhys. Lett."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"093904","DOI":"10.1103\/PhysRevLett.92.093904","article-title":"Gap solitons in waveguide arrays","volume":"92","author":"Mandelik","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_73","first-page":"110","article-title":"Unidirectional flow of the discrete dark solitons and excitation of the discrete X-waves in PT-symmetric optical waveguide arrays","volume":"74","author":"Gao","year":"2022","journal-title":"Rom. Rep. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"1422","DOI":"10.1016\/j.physd.2008.10.001","article-title":"Vortex solutions of the discrete Gross\u2013Pitaevskii equation starting from the anti-continuum limit","volume":"238","author":"Cuevas","year":"2009","journal-title":"Phys. D"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1134\/1.558526","article-title":"Stability of strongly localized excitations in discrete media with cubic nonlinearity","volume":"86","author":"Darmanyan","year":"1998","journal-title":"J. Exp. Theor. Phys."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"036604","DOI":"10.1103\/PhysRevE.63.036604","article-title":"Stability of multiple pulses in discrete systems","volume":"63","author":"Kapitula","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physd.2005.07.021","article-title":"Stability of discrete solitons in nonlinear Schr\u00f6dinger lattices","volume":"212","author":"Pelinovsky","year":"2005","journal-title":"Phys. D"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"9615","DOI":"10.1088\/0305-4470\/34\/45\/302","article-title":"Bound states of two-dimensional solitons in the discrete nonlinear Schr\u00f6dinger equation","volume":"34","author":"Kevrekidis","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"7246","DOI":"10.1103\/PhysRevE.56.7246","article-title":"Resonance in the collision of two discrete intrinsic localized excitations","volume":"56","author":"Cai","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"056603","DOI":"10.1103\/PhysRevE.68.056603","article-title":"Two-soliton collisions in a near-integrable lattice system","volume":"68","author":"Dmitriev","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"043841","DOI":"10.1103\/PhysRevA.86.043841","article-title":"Propagation and collisions of semi-discrete solitons in arrayed and stacked waveguides","volume":"86","author":"Blit","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"062210","DOI":"10.1103\/PhysRevE.97.062210","article-title":"Nonlinearity-induced localization in a periodically driven semidiscrete system","volume":"97","author":"Driben","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"133901","DOI":"10.1103\/PhysRevLett.123.133901","article-title":"Semidiscrete quantum droplets and vortices","volume":"123","author":"Zhang","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"2000082","DOI":"10.1002\/adpr.202000082","article-title":"Semidiscrete vortex solitons","volume":"2","author":"Xu","year":"2021","journal-title":"Adv. Photonics Res."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"123903","DOI":"10.1103\/PhysRevLett.92.123903","article-title":"Observation of discrete vortex solitons in optically induced photonic lattices","volume":"92","author":"Neshev","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"123904","DOI":"10.1103\/PhysRevLett.92.123904","article-title":"Observation of vortex-ring \u201cdiscrete\u201d solitons in 2D photonic lattices","volume":"92","author":"Fleischer","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/0370-1573(92)90116-H","article-title":"The Fermi-Pasta-Ulam problem\u2014Paradox turns discovery","volume":"213","author":"Ford","year":"1992","journal-title":"Phys. Rep."},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Braun, O.M., and Kivshar, Y.S. (2004). The Frenkel\u2013Kontorova Model: Concepts, Methods, and Applications, Springer.","DOI":"10.1007\/978-3-662-10331-9"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"29679","DOI":"10.1364\/OE.22.029679","article-title":"Discrete solitons and vortices on two-dimensional lattices of PT-symmetric couplers","volume":"22","author":"Chen","year":"2014","journal-title":"Opt. Exp."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1088\/0951-7715\/12\/3\/314","article-title":"Excitation thresholds for nonlinear localized modes on lattices","volume":"12","author":"Weinstein","year":"1999","journal-title":"Nonlinearity"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.physd.2008.10.002","article-title":"Multistable solitons in higher-dimensional cubic\u2013quintic nonlinear Schr\u00f6dinger lattices","volume":"238","author":"Chong","year":"2009","journal-title":"Phys. D"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"026601","DOI":"10.1103\/PhysRevE.64.026601","article-title":"Discrete vortex solitons","volume":"64","author":"Malomed","year":"2001","journal-title":"Phys. Rev. E"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.physd.2005.09.015","article-title":"Persistence and stability of discrete vortices in nonlinear Schr\u00f6dinger lattices","volume":"212","author":"Pelinovsky","year":"2005","journal-title":"Phys. D"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1103\/PhysRevLett.13.479","article-title":"Self-Trapping of Optical Beams","volume":"13","author":"Chiao","year":"1964","journal-title":"Phys. Rev. Lett."},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"2277","DOI":"10.1080\/09500349214552301","article-title":"The theory of spiral laser beams in nonlinear media","volume":"39","author":"Kruglov","year":"1992","journal-title":"J. Mod. Opt."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.physd.2019.04.009","article-title":"(INVITED) Vortex solitons: Old results and new perspectives","volume":"399","author":"Malomed","year":"2019","journal-title":"Phys. D"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"056612","DOI":"10.1103\/PhysRevE.70.056612","article-title":"Stable higher-order vortices and quasivortices in the discrete nonlinear Schr\u00f6dinger equation","volume":"70","author":"Kevrekidis","year":"2004","journal-title":"Phys. Rev. E"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"046613","DOI":"10.1103\/PhysRevE.72.046613","article-title":"Discrete solitons and vortices on anisotropic lattices","volume":"72","author":"Kevrekidis","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"2948","DOI":"10.1103\/PhysRevLett.78.2948","article-title":"Self-trapping of an optical vortex by use of the bulk photovoltaic effect","volume":"78","author":"Chen","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"1751","DOI":"10.1364\/OL.22.001751","article-title":"Steady-state vortex-screening solitons formed in biased photorefractive media","volume":"22","author":"Chen","year":"1997","journal-title":"Opt. Lett."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"2456","DOI":"10.1364\/OL.31.002456","article-title":"Self-trapping and flipping of double-charged vortices in optically induced photonic lattices","volume":"31","author":"Bezryadina","year":"2006","journal-title":"Opt. Lett."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"043821","DOI":"10.1103\/PhysRevA.79.043821","article-title":"Observation of double-charge discrete vortex solitons in hexagonal photonic lattices","volume":"79","author":"Terhalle","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"7928","DOI":"10.1103\/PhysRevE.58.7928","article-title":"Potential of interaction between two- and three-dimensional solitons","volume":"58","author":"Malomed","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_104","first-page":"203","article-title":"Gap solitons","volume":"XXXIII","author":"Sipe","year":"1994","journal-title":"Prog. Opt."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1142\/S0217984904007190","article-title":"Theory of nonlinear matter waves in optical lattices","volume":"18","author":"Brazhnyi","year":"2004","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"230401","DOI":"10.1103\/PhysRevLett.92.230401","article-title":"Bright Bose\u2013Einstein gap solitons of atoms with repulsive interaction","volume":"92","author":"Eiermann","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1038\/nphys438","article-title":"Dispersionless slow light using gap solitons","volume":"2","author":"Mok","year":"2006","journal-title":"Nat. Phys."},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"023124","DOI":"10.1063\/1.4881678","article-title":"Localized modes in mini-gaps opened by periodically modulated intersite coupling in two-dimensional nonlinear lattices","volume":"24","author":"Maluckov","year":"2014","journal-title":"Chaos"},{"key":"ref_109","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_110","doi-asserted-by":"crossref","first-page":"046608","DOI":"10.1103\/PhysRevE.76.046608","article-title":"Two-dimensional discrete solitons in rotating lattices","volume":"76","author":"Cuevas","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_111","first-page":"97","article-title":"Spontaneous symmetry breaking in nonlinear systems: An overview and a simple model","volume":"Volume 173","author":"Tlidi","year":"2016","journal-title":"Nonlinear Dynamics: Materials, Theory and Experiments"},{"key":"ref_112","doi-asserted-by":"crossref","unstructured":"Iooss, G., and Joseph, D.D. (1980). Elementary Stability Bifurcation Theory, Springer.","DOI":"10.1007\/978-1-4684-9336-8"},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"036607","DOI":"10.1103\/PhysRevE.74.036607","article-title":"Discrete solitons and vortices in the two-dimensional Salerno model with competing nonlinearities","volume":"74","author":"Malomed","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1038\/nature09887","article-title":"Spin-orbit-coupled Bose\u2013Einstein condensates","volume":"471","author":"Lin","year":"2011","journal-title":"Nature"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1038\/nature11841","article-title":"Spin-orbit coupling in quantum gases","volume":"494","author":"Galitski","year":"2013","journal-title":"Nature"},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"126401","DOI":"10.1088\/0034-4885\/77\/12\/126401","article-title":"Light-induced gauge fields for ultracold atoms","volume":"77","author":"Goldman","year":"2014","journal-title":"Rep. Prog. Phys."},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"026001","DOI":"10.1088\/0034-4885\/78\/2\/026001","article-title":"Degenerate quantum gases with spin\u2013orbit coupling","volume":"78","author":"Zhai","year":"2015","journal-title":"Rep. Prog. Phys."},{"key":"ref_118","doi-asserted-by":"crossref","first-page":"36001","DOI":"10.1209\/0295-5075\/122\/36001","article-title":"Creating solitons by means of spin\u2013orbit coupling","volume":"122","author":"Malomed","year":"2018","journal-title":"EPL"},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"032920","DOI":"10.1103\/PhysRevE.89.032920","article-title":"Creation of two-dimensional composite solitons in spin\u2013orbit-coupled self-attractive Bose\u2013Einstein condensates in free space","volume":"89","author":"Sakaguchi","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"062922","DOI":"10.1103\/PhysRevE.90.062922","article-title":"Discrete and continuum composite solitons in Bose\u2013Einstein condensates with the Rashba spin\u2013orbit coupling in one and two dimensions","volume":"90","author":"Sakaguchi","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_121","doi-asserted-by":"crossref","unstructured":"Pelinovsky, D. (2011). Localization in Periodic Potentials: From Schr\u00f6dinger Operators to the Gross-Pitaevskii Equation, Cambridge University Press.","DOI":"10.1017\/CBO9780511997754"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"080403","DOI":"10.1103\/PhysRevLett.93.080403","article-title":"Three-dimensional solitary waves and vortices in a discrete nonlinear Schr\u00f6dinger lattice","volume":"93","author":"Kevrekidis","year":"2004","journal-title":"Phys. Rev. Lett."},{"key":"ref_123","doi-asserted-by":"crossref","first-page":"203901","DOI":"10.1103\/PhysRevLett.94.203901","article-title":"Three-dimensional nonlinear lattices: From oblique vortices and octupoles to discrete diamonds and vortex cubes","volume":"94","author":"Kevrekidis","year":"2005","journal-title":"Phys. Rev. Lett."},{"key":"ref_124","doi-asserted-by":"crossref","first-page":"3934","DOI":"10.1103\/PhysRevLett.86.3934","article-title":"Creating vortex rings and three-dimensional skyrmions in Bose\u2013Einstein condensates","volume":"86","author":"Ruostekoski","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_125","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1038\/35082010","article-title":"Skyrmions in a ferromagnetic Bose\u2013Einstein condensate","volume":"411","author":"Khawaja","year":"2001","journal-title":"Nature"},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"080401","DOI":"10.1103\/PhysRevLett.88.080401","article-title":"Stable skyrmions in two-component Bose\u2013Einstein condensates","volume":"88","author":"Battye","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"026603","DOI":"10.1103\/PhysRevE.75.026603","article-title":"Skyrmion-like states in two- and three-dimensional dynamical lattices","volume":"75","author":"Kevrekidis","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1088\/0951-7715\/11\/4\/002","article-title":"Skyrmions and domain walls in (2 + 1) dimensions","volume":"11","author":"Kudryavtsevy","year":"1998","journal-title":"Nonlinearity"},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"1489","DOI":"10.1088\/0951-7715\/12\/6\/303","article-title":"The baby Skyrme models and their multi-skyrmions","volume":"12","author":"Weidig","year":"1999","journal-title":"Nonlinearity"},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1364\/OL.19.000329","article-title":"Multidimensional solitons in fiber arrays","volume":"19","author":"Aceves","year":"1994","journal-title":"Opt. Lett."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"263901","DOI":"10.1103\/PhysRevLett.105.263901","article-title":"Three-dimensional light bullets in arrays of waveguides","volume":"105","author":"Minardi","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_132","first-page":"041031","article-title":"Observation of discrete, vortex light bullets","volume":"3","author":"Eilenberger","year":"2013","journal-title":"Phys. Rev. X"},{"key":"ref_133","doi-asserted-by":"crossref","first-page":"38343","DOI":"10.1364\/OE.506130","article-title":"Semidiscrete optical vortex droplets in quasi-phase-matched photonic crystals","volume":"31","author":"Xu","year":"2023","journal-title":"Opt. Exp."},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1364\/OL.31.001097","article-title":"Semi-discrete composite solitons in arrays of quadratically nonlinear waveguides","volume":"31","author":"Panoiu","year":"2006","journal-title":"Opt. Lett."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1088\/0034-4885\/70\/6\/R03","article-title":"Making sense of non-Hermitian Hamiltonians","volume":"70","author":"Bender","year":"2007","journal-title":"Rep. Prog. Phys."},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"103904","DOI":"10.1103\/PhysRevLett.100.103904","article-title":"Beam dynamics in PT symmetric optical lattices","volume":"100","author":"Makris","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1038\/nphys1515","article-title":"Observation of parity\u2013time symmetry in optics","volume":"6","author":"Makris","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"035002","DOI":"10.1103\/RevModPhys.88.035002","article-title":"Nonlinear waves in PT-symmetric systems","volume":"88","author":"Konotop","year":"2016","journal-title":"Rev. Mod. Phys."},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1002\/lpor.201500227","article-title":"Nonlinear switching and solitons in PT-symmetric photonic systems","volume":"10","author":"Suchkov","year":"2016","journal-title":"Laser Phot. Rev."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"4323","DOI":"10.1364\/OL.36.004323","article-title":"Stability of solitons in parity-time-symmetric couplers","volume":"36","author":"Driben","year":"2011","journal-title":"Opt. Lett."},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"063837","DOI":"10.1103\/PhysRevA.85.063837","article-title":"Optical solitons in PT-symmetric nonlinear couplers with gain and loss","volume":"85","author":"Alexeeva","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_142","doi-asserted-by":"crossref","first-page":"062904","DOI":"10.1103\/PhysRevE.88.062904","article-title":"Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic\u2013quintic nonlinearity","volume":"88","author":"Burlak","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_143","doi-asserted-by":"crossref","first-page":"56006","DOI":"10.1209\/0295-5075\/100\/56006","article-title":"Discrete solitons in PT-symmetric lattices","volume":"100","author":"Konotop","year":"2012","journal-title":"Europhys. Lett."},{"key":"ref_144","doi-asserted-by":"crossref","first-page":"3917","DOI":"10.1364\/OE.21.003917","article-title":"Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a PT-symmetry","volume":"21","author":"Huang","year":"2013","journal-title":"Opt. Exp."},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1364\/OL.38.000371","article-title":"Discrete vortex solitons and parity time symmetry","volume":"38","author":"Leykam","year":"2013","journal-title":"Opt. Lett."},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"085204","DOI":"10.1088\/1751-8113\/47\/8\/085204","article-title":"Nonlinear modes in a generalized PT-symmetric discrete nonlinear Schr\u00f6dinger equation","volume":"47","author":"Pelinovsky","year":"2014","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_147","doi-asserted-by":"crossref","first-page":"033207","DOI":"10.1103\/PhysRevE.91.033207","article-title":"Staggered parity-time-symmetric ladders with cubic nonlinearity","volume":"91","author":"Kevrekidis","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"7782","DOI":"10.1038\/ncomms8782","article-title":"Observation of optical solitons in PT-symmetric lattices","volume":"6","author":"Wimmer","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"065301","DOI":"10.1088\/0953-4075\/48\/6\/065301","article-title":"Composite localized modes in discretized spin\u2013orbit-coupled Bose\u2013Einstein condensates","volume":"48","author":"Petrovixcx","year":"2015","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1002\/lapl.200810062","article-title":"Nonlinear Tamm states and surface effects in periodic photonic structures","volume":"5","author":"Kivshar","year":"2008","journal-title":"Laser Phys. Lett."},{"key":"ref_151","doi-asserted-by":"crossref","first-page":"1780","DOI":"10.1038\/s41598-017-01646-y","article-title":"Edge solitons in kagome lattice","volume":"7","author":"Gulevich","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_152","doi-asserted-by":"crossref","first-page":"1228","DOI":"10.1364\/OPTICA.3.001228","article-title":"Modulational instability and solitary waves in polariton topological insulators","volume":"3","author":"Kartashov","year":"2016","journal-title":"Optica"},{"key":"ref_153","doi-asserted-by":"crossref","first-page":"126835","DOI":"10.1016\/j.physleta.2020.126835","article-title":"The two-dimensional fractional discrete nonlinear Schr\u00f6dinger equation","volume":"384","author":"Molina","year":"2020","journal-title":"Phys. Lett. A"},{"key":"ref_154","doi-asserted-by":"crossref","first-page":"7347","DOI":"10.1103\/PhysRevA.46.R7347","article-title":"Dynamics of the globally coupled complex Ginzburg-Landau equation","volume":"46","author":"Hakim","year":"1992","journal-title":"Phys. Rev. A"},{"key":"ref_155","doi-asserted-by":"crossref","first-page":"026606","DOI":"10.1103\/PhysRevE.67.026606","article-title":"Discrete Ginzburg-Landau solitons","volume":"67","author":"Efremidis","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_156","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0030-4018(03)01457-3","article-title":"Exact localized and periodic solutions of the discrete complex Ginzburg-Landau equation","volume":"221","author":"Maruno","year":"2003","journal-title":"Opt. Commun."},{"key":"ref_157","doi-asserted-by":"crossref","first-page":"043839","DOI":"10.1103\/PhysRevA.76.043839","article-title":"Two-dimensional discrete Ginzburg-Landau solitons","volume":"76","author":"Efremidis","year":"2007","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/2\/137\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:53:26Z","timestamp":1760104406000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/2\/137"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,2]]},"references-count":157,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,2]]}},"alternative-id":["e26020137"],"URL":"https:\/\/doi.org\/10.3390\/e26020137","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,2,2]]}}}