Semidiscrete Integrable Nonlinear Schr¨odinger System with Background-Dependent Intersite Interaction

  • O. O. Vakhnenko Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
Keywords: -


We summarize the most featured items characterizing the semi-discrete nonlinear Schr¨odinger system with background-controlled inter-site resonant coupling. The system is shown to be integrable in the Lax sense that make it possible to obtain its soliton solutions in the framework of properly parameterized dressing procedure based on the Darboux transformation. On the other hand the system integrability inspires an infinite hierarchy of local conservation laws some of which were found explicitly in the framework of generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem and it permits 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 inter-site resonant coupling and as a consequence it establishes the triangularlattice-ribbon spatial arrangement of location sites for the basic field excitations. Adjusting the background parameter we are able to switch over the system dynamics between two essentially different regimes separated by the critical point. The system criticality against the background parameter is manifested both indirectly by the auxiliary linear spectral problem and directly by the nonlinear dynamical equations themselves. The physical implications of system criticality become evident after the rather sophisticated canonization procedure of basic field variables. There are two variants of system standardization 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 support the bright excitations in both subsystems, while in the over-critical region one of subsystems converts into the subsystem of dark excitations.


M.J. Ablowitz, J.F. Ladik. Nonlinear differential-difference equations. J. Math. Phys. 16, 598–603 (1975).

M.J. Ablowitz, J.F. Ladik. Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17, 1011–1018 (1976).

M.J. Ablowitz, Y. Ohta, A.D. Trubatch. On discretizations of the vector nonlinear Schr¨odinger equation. Phys. Lett. A 253, 287–304 (1999).

M.J. Ablowitz, B. Prinari, A.D. Trubatch. Discrete and Continuous Nonlinear Schr¨odinger Systems (Cambridge University Press, New York, 2004).

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, 1711–1758 (2007).

V.E. Adler, V.V. Postnikov. On vector analogs of the modified Volterra lattice. J. Phys. A: Math. Theor. 41, 455203 (2008).

G. Biondini, D. Kraus. Inverse scattering transform for the defocusing Manakov system with nonzero boundary condirions. SIAM J. Math. Anal. 47, 706–757 (2015).

G. Biondini, D.K. Kraus, B. Prinari. The three-component defocusing nonlinear Schr¨odinger equation with nonzero boundary conditions. Commun. Math. Phys. 348, 475–533 (2016).

L.S. Brizhik, B.M.A.G. Piette, W.J. Zakrzewski. Donor-acceptor electron transport mediated by solitons. Phys. Rev. E 90 052915 (2014).

R.K. Bullough, N.M. Bogoliubov, A.V. Rybin, G.G. Varzugin, J. Timonen. Solitons of q-deformed quantum lattices and the quantum soliton. J. Phys. A: Math. Gen. 34, 10463–10474 (2001).

P.J. Caudrey. The inverse problem for a general N × N spectral equation. Physica D 6, 51–66 (1982).

P.J. Caudrey. Differential and discrete spectral problems and their inverses. In: Wave Phenomena: Modern Theory and Applications, North-Holland Mathematics Studies 97 (Elsevier, Amsterdam, 1984), pp. 221–232.

A.R. Chowdhury, G. Mahato. A Darboux-B¨acklund transformation associated with a discrete nonlinear Sch¨odinger equation. Lett. Math. Phys. 7, 313–317 (1983).

D.N. Christodoulides, R.I. Joseph. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 13, 794–796 (1988).

G. Darboux. Sur le probl`eme de Pfaff. Bull. Sci. Math. Astron. 2 s´erie 6, 14–36 (1882).

G. Darboux, Sur le probl`eme de Pfaff, Bull. Sci. Math. Astron. 2 s´erie 6, 49–68 (1882).

А.С. Давыдов. Теория молекулярных экситонов (Наука, Москва, 1968); A.S. Davydov. Theory of Molecular Excitons (Plenum Press, New York–London, 1971).

А.С. Давыдов, A.A. Еремко, А.И. Сергиенко. Солитоны в a-спиральных белковых молекулах. Укр. Физ. Журн. 23, 983–993 (1978).

А.С. Давыдов. Солитоны в молекулярных системах (Наукова Думка, Київ, 1984); A.S. Davydov. Solitons in Molecular Systems (Kluwer Academic, Dordrecht, 1991).

E.V. Doktorov, S.B. Leble. A Dressing Method in Mathematical Physics (Springer, Dordrecht, 2007).

Б.А. Дубровин, С.П. Новиков, А.Ф. Фоменко. Современная геометрия. Методы и приложения (Наука, Москва, 1986); B.A. Dubrovin, A.F. Fomenko, S.P. Novikov. Modern Geometry. Methods and Applications (Springer-Verlag, Berlin, 1984).

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).

I.L. Garanovich, S. Longhi, A.A. Sukhorukov, Yu.S. Kivshar. Light propagation and localization in modulated photonic lattices and waveguides. Phys. Rep. 518, 1–79 (2012).

В.С. Герджиков, М.И. Иванов. Гамильтонова структура многокомпонентных разностных нелинейных уравнений Шредингера. Теор. Мат. Физ. 52, 89–104 (1982); V.S. Gerdzhikov, M.I. Ivanov. Hamiltonian structure of multicomponent nonliner Schr¨odinger equations in difference form. Theor. Math. Phys. 52, 676–685 (1982).

C.H. Gu, H.S. Hu, Z.X. Zhou. Darboux Transformations in Integrable Systems. Theory and their Applications to Geometry (Kluwer Academic Publishers, Boston–Dordrecht–London, 2005).

L. Jiao, L. Zhang, X. Wang, G. Diankov, H. Dai. Narrow graphene nanoribbons from carbon nanotubes. Nature 458, 877–880 (2009).

F. Kako, N. Mugibayashi. Complete integrability of general nonlinear differential-difference equations solvable by the inverse method. II. Progr. Theor. Phys. 61, 776–790 (1979).

Yu.S. Kivshar, B. Luther-Davies. Dark optical solitons: Physics and applications. Phys. Rep. 298, 81–197 (1998).

K. Konno, H. Sanuki, Y.H. Ichikawa. Conservation laws of nonlinear-evolution equations. Progr. Theor. Phys. 52, 886–889 (1974).

V.V. Konotop, O.A. Chubykalo, L. V´azquez. Dynamics and interaction of solitons on an integrable inhomogeneous lattice. Phys. Rev. E 48, 563–568 (1993).

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, 872–876 (2009).

P.P. Kulish. Quantum difference nonlinear Schr¨odinger equation. Lett. Math. Phys. 5, 191–197 (1981).

V. Kuznetsov, E. Sklyanin. B¨acklund transformation for the BC-type Toda lattice. SIGMA 3, 080 (2007).

С.Б. Лебле, М.А. Салль. Преобразование Дарбу для дискретного аналога уравнений Силина-Тихончука. Доклады АН СССР 284, 110–114 (1985); S.B. Leble, M.A. Sall’. The Darboux transformation for the discrete analog of the Silin-Tikhonchuk equations. Sov. Phys.–Doklady 30, 760–762 (1985).

S.B. Leble. Nonlinear Waves and Waveguides with Stratification (Springer-Verlag, Berlin–Heidelberg, 1991).

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).

P. Marqui´e, J.M. Bilbault, M. Remoissenet. Nonlinear Schr¨odinger models and modulational instability in real electrical lattices. Physica D 87, 371–374 (1995).

B.M. Maschke, A.J. Van Der Schaft, P.C. Breedveld. An intrinsic Hamiltonian formulation of network dynamics: Non-standard Poisson structures and gyrators. J. Franklin Inst. 329, 923–966 (1992).

V.B. Matveev. Darboux transformation and the explicit solutions of differential-difference and difference-difference evolution equations. I. Lett. Math. Phys. 3, 217–222 (1979).

V.B. Matveev, M.A. Salle. Differential-difference evolution equations. II (Darboux transformation for the Toda lattice). Lett. Math. Phys. 3, 425–429 (1979).

V.B. Matveev, M.A. Salle. Darboux Transformations and Solitons (Springer-Verlag, Berlin – Heidelberg, 1991).

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, 126–132 (2014).

G. Neugebauer, R. Meinel. General N-soliton solution of the AKNS class on arbitrary background. Phys. Lett. A 100, 467–470 (1984).

A.C. Newell. Solitons in Mathematics and Physics (SIAM Press, Philadelphia, 1985).

R. Peierls. Zur theorie des diamagnetismus von leitungselektronen. Z. Phys. 80, 763–791 (1933).

A. Rybin, J. Timonen, G. Varzugin, R.K. Bullough. q-deformed solitons and quantum solitons of the Maxwell-Bloch lattice. J. Phys. A: Math. Gen. 34, 157–164 (2001).

R. Scharf, A.R. Bishop. Properties of the nonlinear Schr¨odinger equation on a lattice. Phys. Rev. A 61, 6535–6544 (1991).

A.C. Scott. Dynamics of Davydov solitons. Phys. Rev. A 26, 578–595 (1982).

В.П. Силин, В.Т. Тихончук. Параметрическая турбулентность и Черенковское излучение электронов в пространственно неднородной плазме. Журн. Експ. Теор. Физ. 81, 2039–2051 (1981); V.P. Silin, V.T. Tikhonchuk. Parametric turbulence and Cherenkov heating of electrons in a spatially inhomogeneous plasma. Sov. Phys.–JETP 54, 1075–1082 (1981).

Л.А. Тахтаджян, Л.Д. Фаддеев. Гамильтонов подход в теории солитонов (Наука,Москва, 1986); L.D. Faddeev, L.A. Takhtajan. Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).

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, 2425–2437 (2007).

T. Tsuchida, H. Ujino, M. Wadati. Integrable semi-discretization of the coupled modified KdV equations. J. Math. Phys. 39, 4785–4813 (1998).

T. Tsuchida, H. Ujino, M. Wadati. Integrable semi-discretization of the coupled nonlinear Schr¨odinger equations. J. Phys. A: Math. Gen. 32, 2239–2262 (1999).

G.-Z. Tu. A trace identity and its applications to the theory of discrete integrable systems. J. Phys. A: Math. Gen. 23, 3903–3922 (1990).

О.О. Вахненко. Нова повнiстю iнтеґровра дискретизацiя нелiнiйного рiвняння Шрьодiнґера. Укр. Фiз. Журн. 40, 118–122 (1995).

O.O. Vakhnenko, V.O. Vakhnenko. Physically corrected Ablowitz-Ladik model and its application to the Peierls-Nabarro problem. Phys. Lett. A 196, 307–312 (1995).

O.O. Vakhnenko. Nonlinear beating excitations on ladder lattice. J. Phys. A: Math. Gen. 32, 5735–5748 (1999).

O.O. Vakhnenko, M.J. Velgakis. Transverse and longitudinal dynamics of nonlinear intramolecular excitations on multileg ladder lattices. Phys. Rev. E 61, 7110–7120 (2000).

O.O. Vakhnenko. Solitons on a zigzag-runged ladder lattice. Phys. Rev. E 64, 067601 (2001).

O.O. Vakhnenko. Solitons in parametrically driven discrete nonlinear Schr¨odinger systems with the exploding range of intersite interactions. J. Math. Phys. 43, 2587–2605 (2002).

O.O. Vakhnenko. Three component nonlinear dynamical system generated by the new third-order discrete spectral operator. J. Phys. A: Math. Gen. 36 (2003) 5405–5430.

O.O. Vakhnenko. A discrete nonlinear model of three coupled dynamical fields integrable by the Caudrey method. Ukr. J. Phys. 48, 653–666 (2003).

O.O. Vakhnenko. Integrable nonlinear ladder system with background-controlled intersite resonant coupling. J. Phys. A: Math. Gen. 39, 11013–11027 (2006).

O.O. Vakhnenko. Enigma of probability amplitudes in Hamiltonian formulation of integrable semidiscrete nonlinear Schr¨odinger systems. Phys. Rev. E 77, 026604 (2008).

O.O. Vakhnenko. Inverse scattering transform for the nonlinear Schr¨odinger system on a zigzag-runged ladder lattice. J. Math. Phys. 51, 103518 (2010).

O.O. Vakhnenko. Semidiscrete integrable nonlinear systems generated by the new fourth-order spectral operator. Local conservation laws. J. Nonlin. Math. Phys. 18, 401–414 (2011).

O.O. Vakhnenko. Integrable nonlinear Schr¨odinger system on a triangular-lattice ribbon. J. Phys. Soc. Japan 84, 014003 (2015).

O.O. Vakhnenko. Nonlinear integrable model of Frenkel-like excitations on a ribbon of triangular lattice. J. Math. Phys. 56, 033505 (2015).

O.O. Vakhnenko. Coupling-governed metamorphoses of the integrable nonlinear Schr¨odinger system on a triangular-lattice ribbon. Phys. Lett. A 380, 2069–2074 (2016).

O.O. Vakhnenko. Asymmetric canonicalization of the integrable nonlinear Schr¨odinger system on a triangular-lattice ribbon. Appl. Math. Lett. 64, 81–86 (2017).

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).

O.O. Vakhnenko. Distinctive features of the integrable nonlinear Schr¨odinger system on a ribbon of triangular lattice. Ukr. J. Phys. 62, 271–282 (2017).

V.O. Vakhnenko, E.J. Parkes. The singular solutions of a nonlinear evolution equation taking continuous part of the spectral data into account in inverse scattering method. Chaos, Solitons and Fractals 45, 846–852 (2012).

V.E. Vekslerchik, V.V. Konotop. Discrete nonlinear Schr¨odinger equation under non-vanishing boundary conditions. Inverse Problems 8, 889–909 (1992).

M. Wadati, H. Sanuki, K. Konno. Relationships among inverse method, B¨acklund transformation and an infinite number of conservation laws. Progr. Theor. Phys. 53, 419–436 (1975).

X.-Y. Wen. New hierarchies of integrable lattice equations and associated properties: Darboux transformation, conservation laws and integrable coupling. Rep. Math. Phys. 67, 259–277 (1975).

X.-X. Xu. Darboux transformation of a coupled lattice soliton equation. Phys. Lett. A 362, 205–211 (2007).

В.Е. Захаров, Е.А. Кузнецов. Гамильтоновский формализм для нелинейных волн. Усп. Физ. Наук 167, 1137–1167 (1997); V.E. Zakharov, E.A. Kuznetsov. Hamiltonian formalism for nonlinear waves. Phys.–Uspekhi 40, 1087–1116 (1997).

J.M. Ziman. Models of disorder. The Theoretical Physics of Homogeneously Disordered Systems (Cambridge University Press, Cambridge, 1979).

How to Cite
Vakhnenko, O. (2019). Semidiscrete Integrable Nonlinear Schr¨odinger System with Background-Dependent Intersite Interaction. Ukrainian Journal of Physics, 12(1), 3. Retrieved from