Semidiscrete Integrable Nonlinear Schrӧdinger System with Background-Controlled Intersite Resonant Coupling. Short Summary of Key Properties
DOI:
https://doi.org/10.15407/ujpe63.3.220Keywords:
nonlinear lattice, integrable system, soliton, conservation laws, symmetry breaking, canonical field variablesAbstract
The most featured items characterizing the semidiscrete nonlinear Schr¨odinger system with background-controlled intersite resonant coupling are summarized. The system is shown to be integrable in the Lax sense that makes it possible to obtain its soliton solutions in the framework of a properly parametrized dressing procedure based on the Darboux transformation accompanied by the implicit form of B¨acklund transformation. In addition, the system integrability inspires an infinite hierarchy of local conservation laws, some of which were found explicitly in the framework of the generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem, and its dynamics is embedded into the Hamiltonian formulation accompanied by the highly nonstandard Poisson structure. The nonzero background level of concomitant fields mediates the appearance of an additional type of the intersite resonant coupling. As a consequence, it establishes the triangular-lattice-ribbon spatial arrangement of location sites for the basic field excitations. At tuning the main background parameter, we are able to switch system’s dynamics between two essentially different regimes separated by the critical point. The physical implications of system’s criticality become evident after a rather sophisticated procedure of canonization of basic field variables. There are two variants to standardize the system equal in their rights. Each variant is realizable in the form of two nonequivalent canonical subsystems. The broken symmetry between canonical subsystems gives rise to the crossover effect in the nature of excited states. Thus, in the under-critical region, the system supports the bright excitations in both subsystems; while, in the over-critical region, one of the subsystems converts into the subsystem of dark excitations.
References
<li>M.J. Ablowitz, J.F. Ladik. Nonlinear differential-difference equations. J. Math. Phys. 16, 598 (1975).
<a href="https://doi.org/10.1063/1.522558">https://doi.org/10.1063/1.522558</a>
</li>
<li>M.J. Ablowitz, J.F. Ladik. Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17, 1011 (1976).
<a href="https://doi.org/10.1063/1.523009">https://doi.org/10.1063/1.523009</a>
</li>
<li>M.J. Ablowitz, Y. Ohta, A.D. Trubatch. On discretizations of the vector nonlinear Schr?odinger equation. Phys. Lett. A 253, 287 (1999).
<a href="https://doi.org/10.1016/S0375-9601(99)00048-1">https://doi.org/10.1016/S0375-9601(99)00048-1</a>
</li>
<li>M.J. Ablowitz, B. Prinari, A.D. Trubatch. Discrete and Continuous Nonlinear Schr?odinger Systems (Cambridge Univ. Press, 2004).
</li>
<li>M.J. Ablowitz, G. Biondini, B. Prinari. Inverse scattering transform for the integrable discrete nonlinear Schr?odinger equation with nonvanishing boundary conditions. Inverse Problems 23, 1711 (2007).
<a href="https://doi.org/10.1088/0266-5611/23/4/021">https://doi.org/10.1088/0266-5611/23/4/021</a>
</li>
<li>L.S. Brizhik, B.M.A.G. Piette, W.J. Zakrzewski. Donor-acceptor electron transport mediated by solitons. Phys. Rev. E 90, 052915 (2014).
<a href="https://doi.org/10.1103/PhysRevE.90.052915">https://doi.org/10.1103/PhysRevE.90.052915</a>
</li>
<li>D.N. Christodoulides, R.I. Joseph. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 13, 794 (1988).
<a href="https://doi.org/10.1364/OL.13.000794">https://doi.org/10.1364/OL.13.000794</a>
</li>
<li>A.S. Davydov. Theory of Molecular Excitons (Plenum Press, 1971).
<a href="https://doi.org/10.1007/978-1-4899-5169-4">https://doi.org/10.1007/978-1-4899-5169-4</a>
</li>
<li>A.S. Davydov, A.A. Eremko, A.I. Sergienko. Solitons in a-helix protein molecules. Ukr. J. Phys. 23, 983 (1978).
</li>
<li> A.S. Davydov. Solitons in Molecular Systems (Kluwer Academic, 1991).
<a href="https://doi.org/10.1007/978-94-011-3340-1">https://doi.org/10.1007/978-94-011-3340-1</a>
</li>
<li> M. Eliashvili, G.I. Japaridze, G. Tsitsishvili, G. Tukhashvili. Edge states in 2D lattices with hopping anisotropy and Chebyshev polynomials. J. Phys. Soc. Japan 83, 044706 (2014).
<a href="https://doi.org/10.7566/JPSJ.83.044706">https://doi.org/10.7566/JPSJ.83.044706</a>
</li>
<li> I.L. Garanovich, S. Longhi, A.A. Sukhorukov, Yu.S. Kivshar. Light propagation and localization in modulated photonic lattices and waveguides. Phys. Rep. 518, 1 (2012).
<a href="https://doi.org/10.1016/j.physrep.2012.03.005">https://doi.org/10.1016/j.physrep.2012.03.005</a>
</li>
<li> V.S. Gerdzhikov, M.I. Ivanov. Hamiltonian structure of multicomponent nonliner Schr?odinger equations in difference form. Theor. Math. Phys. 52, 676 (1982).
<a href="https://doi.org/10.1007/BF01027788">https://doi.org/10.1007/BF01027788</a>
</li>
<li> L. Jiao, L. Zhang, X. Wang, G. Diankov, H. Dai. Narrow graphene nanoribbons from carbon nanotubes. Nature 458, 877 (2009).
<a href="https://doi.org/10.1038/nature07919">https://doi.org/10.1038/nature07919</a>
</li>
<li> Yu.S. Kivshar, B. Luther-Davies. Dark optical solitons: Physics and applications. Phys. Rep. 298, 81 (1998).
<a href="https://doi.org/10.1016/S0370-1573(97)00073-2">https://doi.org/10.1016/S0370-1573(97)00073-2</a>
</li>
<li> D.V. Kosynkin, A.L. Higginbotham, A. Sinitskii, J.R. Lomeda, A. Dimiev, B.K. Price, J.M. Tour. Longitudinal unzipping of carbon nanotubes to form graphene nanoribbons. Nature 458, 872 (2009).
<a href="https://doi.org/10.1038/nature07872">https://doi.org/10.1038/nature07872</a>
</li>
<li> P.P. Kulish. Quantum difference nonlinear Schr?odinger equation. Lett. Math. Phys. 5, 191 (1981).
<a href="https://doi.org/10.1007/BF00420698">https://doi.org/10.1007/BF00420698</a>
</li>
<li> R.K.F. Lee, B.J. Cox, J.M. Hill. An exact polyhedral model for boron nanotubes. J. Phys. A: Math. Theor. 42, 065204 (2009).
<a href="https://doi.org/10.1088/1751-8113/42/6/065204">https://doi.org/10.1088/1751-8113/42/6/065204</a>
</li>
<li> P. Marqui’e, J.M. Bilbault, M. Remoissenet. Nonlinear Schr?odinger models and modulational instability in real electrical lattices. Physica D 87, 371 (1995).
<a href="https://doi.org/10.1016/0167-2789(95)00162-W">https://doi.org/10.1016/0167-2789(95)00162-W</a>
</li>
<li> A. Narita, X. Feng, Y. Hernandez, S.A. Jensen, M. Bonn, H. Yang, I.A. Verzhbitskiy, C. Casiraghi, M.R. Hansen, A.H.R. Koch, G. Fytas, O. Ivasenko, B. Li, K.S. Mali, T. Balandina, S. Mahesh, S. De Feyter, K. M?ullen. Synthesis of structurally well-defined and liquid-phase-processable graphene nanoribbons. Nature Chemistry 6, 126 (2014).
<a href="https://doi.org/10.1038/nchem.1819">https://doi.org/10.1038/nchem.1819</a>
</li>
<li> A.C. Newell. Solitons in Mathematics and Physics (SIAM Press, 1985).
<a href="https://doi.org/10.1137/1.9781611970227">https://doi.org/10.1137/1.9781611970227</a>
</li>
<li> R. Peierls. Zur theorie des diamagnetismus von leitungselektronen. Z. Phys. 80, 763 (1933).
<a href="https://doi.org/10.1007/BF01342591">https://doi.org/10.1007/BF01342591</a>
</li>
<li> A.C. Scott. Dynamics of Davydov solitons. Phys. Rev. A 26, 578 (1982).
<a href="https://doi.org/10.1103/PhysRevA.26.578">https://doi.org/10.1103/PhysRevA.26.578</a>
</li>
<li> L.D. Faddeev and L.A. Takhtajan. Hamiltonian Methods in the Theory of Solitons (Springer, 1987).
<a href="https://doi.org/10.1007/978-3-540-69969-9">https://doi.org/10.1007/978-3-540-69969-9</a>
</li>
<li> Y. Tang, J. Cao, X. Liu, Y. Sun. Symplectic methods for the Ablowitz–Ladik discrete nonlinear Schr?odinger equation. J. Phys. A: Math. Theor. 40, 2425 (2007).
<a href="https://doi.org/10.1088/1751-8113/40/10/012">https://doi.org/10.1088/1751-8113/40/10/012</a>
</li>
<li> T. Tsuchida, H. Ujino, M. Wadati. Integrable semidiscretization of the coupled nonlinear Schr?odinger equations. J. Phys. A: Math. Gen. 32, 2239 (1999).
<a href="https://doi.org/10.1088/0305-4470/32/11/016">https://doi.org/10.1088/0305-4470/32/11/016</a>
</li>
<li> O.O. Vakhnenko. The new comlpletely integrable discretization of the nonlinear Schr?odinger equation). Ukr. J. Phys. 40, 118 (1995).
</li>
<li> O.O. Vakhnenko, V.O. Vakhnenko. Physically corrected Ablowitz–Ladik model and its application to the Peierls–Nabarro problem. Phys. Lett. A 196, 307 (1995).
<a href="https://doi.org/10.1016/0375-9601(94)00913-A">https://doi.org/10.1016/0375-9601(94)00913-A</a>
</li>
<li> O.O. Vakhnenko. Nonlinear beating excitations on ladder lattice. J. Phys. A: Math. Gen. 32, 5735 (1999).
<a href="https://doi.org/10.1088/0305-4470/32/30/315">https://doi.org/10.1088/0305-4470/32/30/315</a>
</li>
<li> O.O. Vakhnenko, M.J. Velgakis. Transverse and longitudinal dynamics of nonlinear intramolecular excitations on multileg ladder lattices. Phys. Rev. E 61, 7110 (2000).
<a href="https://doi.org/10.1103/PhysRevE.61.7110">https://doi.org/10.1103/PhysRevE.61.7110</a>
</li>
<li> O.O. Vakhnenko. Solitons on a zigzag-runged ladder lattice. Phys. Rev. E 64, 067601 (2001).
<a href="https://doi.org/10.1103/PhysRevE.64.067601">https://doi.org/10.1103/PhysRevE.64.067601</a>
</li>
<li> O.O. Vakhnenko. Integrable nonlinear ladder system with background-controlled intersite resonant coupling. J. Phys. A: Math. Gen. 39, 11013 (2006).
<a href="https://doi.org/10.1088/0305-4470/39/35/005">https://doi.org/10.1088/0305-4470/39/35/005</a>
</li>
<li> O.O. Vakhnenko. Enigma of probability amplitudes in Hamiltonian formulation of integrable semidiscrete nonlinear Schr?odinger systems. Phys. Rev. E 77, 026604 (2008).
<a href="https://doi.org/10.1103/PhysRevE.77.026604">https://doi.org/10.1103/PhysRevE.77.026604</a>
</li>
<li> O.O. Vakhnenko. Semidiscrete integrable nonlinear systems generated by the new fourth-order spectral operator. Local conservation laws. J. Nonlin. Math. Phys. 18, 401 (2011).
<a href="https://doi.org/10.1142/S1402925111001672">https://doi.org/10.1142/S1402925111001672</a>
</li>
<li> O.O. Vakhnenko. Integrable nonlinear Schr?odinger system on a triangular-lattice ribbon. J. Phys. Soc. Japan 84, 014003 (2015).
<a href="https://doi.org/10.7566/JPSJ.84.014003">https://doi.org/10.7566/JPSJ.84.014003</a>
</li>
<li> O.O. Vakhnenko. Nonlinear integrable model of Frenkel-like excitations on a ribbon of triangular lattice. J. Math. Phys. 56, 033505 (2015).
<a href="https://doi.org/10.1063/1.4914510">https://doi.org/10.1063/1.4914510</a>
</li>
<li> O.O. Vakhnenko. Coupling-governed metamorphoses of the integrable nonlinear Schr?odinger system on a triangular-lattice ribbon. Phys. Lett. A 380, 2069 (2016).
<a href="https://doi.org/10.1016/j.physleta.2016.04.034">https://doi.org/10.1016/j.physleta.2016.04.034</a>
</li>
<li> O.O. Vakhnenko. Asymmetric canonicalization of the integrable nonlinear Schr?odinger system on a triangular-lattice ribbon. Appl. Math. Lett. 64, 81 (2017).
<a href="https://doi.org/10.1016/j.aml.2016.07.013">https://doi.org/10.1016/j.aml.2016.07.013</a>
</li>
<li> O.O. Vakhnenko. Symmetry-broken canonizations of the semi-discrete integrable nonlinear Schr?odinger system with background-controlled intersite coupling. J. Math. Phys. 57, 113504 (2016).
<a href="https://doi.org/10.1063/1.4968244">https://doi.org/10.1063/1.4968244</a>
</li>
<li> O.O. Vakhnenko. Distinctive features of the integrable non-linear Schr?odinger system on a ribbon of triangular lattice. Ukr. J. Phys. 62, 271 (2017).
<a href="https://doi.org/10.15407/ujpe62.03.0271">https://doi.org/10.15407/ujpe62.03.0271</a>
</li>
<li> O.O. Vakhnenko. Semi-discrete integrable nonlinear Schr?odinger system with background-controlled inter-site resonant coupling. J. Nonlin. Math. Phys. 24, 250 (2017).
<a href="https://doi.org/10.1080/14029251.2017.1316011">https://doi.org/10.1080/14029251.2017.1316011</a>
</li>
<li> O.O. Vakhnenko. Semi-discrete integrable nonlinear Schr?odinger system with background-dependent intersite interaction. Ukr. J. Phys. Reviews 12, 3 (2017).
</li>
<li> V.E. Vekslerchik, V.V. Konotop. Discrete nonlinear Schr?odinger equation under non-vanishing boundary conditions. Inverse Problems 8, 889 (1992).
<a href="https://doi.org/10.1088/0266-5611/8/6/007">https://doi.org/10.1088/0266-5611/8/6/007</a>
</li>
<li> J.M. Ziman. Models of Disorder. The Theoretical Physics of Homogeneously Disordered Systems (Cambridge Univ. Press, 1979).
</li></ol>
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