Development and Analysis of novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Parametrically Driven Nonlinear System of Pseudo-Excitations on a Two-Leg Ladder Lattice
DOI:
https://doi.org/10.15407/ujpe69.8.577Keywords:
nonlinear dynamics, integrable system, two-leg ladder lattice, parametric drive, Hamiltonian dynamicsAbstract
Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices, we suggest a novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and evolution operators specified by the properly organized 3 × 3 square matrices. Although the lowest conserved local densities found via the direct recursive method do not prompt us the algebraic structure of system’s Hamiltonian function, but the heuristically substantiated search for the suitable two-stage transformation of prototype field functions to the physically motivated ones has allowed to disclose the physically meaningful nonlinear integrable system with time-dependent longitudinal and transverse inter-site coupling parameters. The time dependencies of inter-site coupling parameters in the transformed system are consistently defined in terms of the accompanying parametric driver formalized by the set of four homogeneous ordinary linear differential equations with the time-dependent coefficients. The physically meaningful parametrically driven nonlinear system permits its concise Hamiltonian formulation with the two pairs of field functions serving as the two pairs of canonically conjugated field amplitudes. The explicit example of oscillatory parametric drive is described in full mathematical details.
References
L.D. Landau, S.I. Pekar. Effective mass of a polyaron. Ukr. J. Phys. 53 (Special Issue), 71 (2008).
N.N. Bogolyubov. On a new form of adiabatic perturbation theory in the problem of particle interaction with a quantum field. Ukr. Mat. Zhurnal 2 (2), 3 (1950).
T. Holstein. Studies of polaron motion: Part I. The molecular-crystal model. Ann. Phys. 8 (3), 325 (1959).
https://doi.org/10.1016/0003-4916(59)90002-8
A.S. Davydov, N.I. Kislukha. Solitary excitons in onedimensional molecular chains. Phys. Stat. Solidi B 59 (2), 465 (1973).
https://doi.org/10.1002/pssb.2220590212
A.S. Davydov, N.I. Kislukha. Solitons in one-dimensional molecular chains. Phys. Stat. Solidi B 75 (2), 735 (1976).
https://doi.org/10.1002/pssb.2220750238
E.G. Wilson. A new theory of acoustic solitary-wave polaron motion. J. Phys. C: Solid State Phys. 16 (35), 6739 (1983).
https://doi.org/10.1088/0022-3719/16/35/008
V.E. Zakharov, A.B. Shabat. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys.-JETP 34 (1), 62 (1972).
S.V. Manakov. Complete integrability and stochastization of discrete dynamical systems. Sov. Phys.-JETP 40 (2), 269 (1975).
O.O. Vakhnenko. Nonlinear beating excitations on ladder lattice. J. Phys. A: Math. Gen. 32 (30), 5735 (1999).
https://doi.org/10.1088/0305-4470/32/30/315
M.J. Ablowitz, J.F. Ladik. Nonlinear differential-difference equations. J. Math. Phys. 16 (3), 598 (1975).
https://doi.org/10.1063/1.522558
M.J. Ablowitz, J.F. Ladik. A nonlinear difference scheme and inverse scattering. Stud. Appl. Math. 55 (3), 213 (1976).
https://doi.org/10.1002/sapm1976553213
M.J. Ablowitz. Lectures on the inverse scattering transform. Stud. Appl. Math. 58 (1), 17 (1978).
https://doi.org/10.1002/sapm197858117
T. Tsuchida, H. Ujino, M. Wadati. Integrable semidiscretization of the coupled nonlinear Schr¨odinger equations. J. Phys. A: Math. Gen. 32 (11), 2239 (1999).
https://doi.org/10.1088/0305-4470/32/11/016
O.O. Vakhnenko, M.J. Velgakis. Transverse and longitudinal dynamics of nonlinear intramolecular excitations on multileg ladder lattices. Phys. Rev. E 61 (6), 7110 (2000).
https://doi.org/10.1103/PhysRevE.61.7110
O.O. Vakhnenko. Inverse scattering transform for the nonlinear Schr¨odinger system on a zigzag-runged ladder lattice. J. Math. Phys. 51 (10), 103518 (2010).
https://doi.org/10.1063/1.3481565
O.O. Vakhnenko. Integrable nonlinear Schr¨odinger system on a lattice with three structural elements in the unit cell. J. Math. Phys. 59 (5), 053504 (2018).
https://doi.org/10.1063/1.4994622
Y.N. Joglekar, C. Thompson, D.D. Scott, G. Vemuria. Optical waveguide arrays: Quantum effects and PT symmetry breaking. Eur. Phys. J. Appl. Phys. 63 (3), 30001 (2013).
https://doi.org/10.1051/epjap/2013130240
Z. Chen, A. Narita, K. M¨ullen. Graphene nanoribbons: On-surface synthesis and integration into electronic devices. Adv. Mater. 32 (45), 2001893 (2020).
https://doi.org/10.1002/adma.202001893
A. Dwivedi, A. Banerjee, B. Bhattacharya. Simultaneous energy harvesting and vibration attenuation in piezoembedded negative stiffness metamaterial. J. Intell. Mater. Syst. Struc. 31 (8), 1 (2020).
https://doi.org/10.1177/1045389X20910261
M. Rothe, Y. Zhao, J. M¨uller, G. Kewes, C.T. Koch, Y. Lu, O. Benson. Self-assembly of plasmonic nanoantenna-waveguide structures for subdiffractional chiral sensing. ACS Nano 15 (1), 351 (2021).
https://doi.org/10.1021/acsnano.0c05240
J.-C. Deinert, D.A. Iranzo, R. P'erez, X. Jia, H.A. Hafez, I. Ilyakov, N. Awari, M. Chen, M. Bawatna, A.N. Ponomaryov, S. Germanskiy, M. Bonn, F.H.L. Koppens, D. Turchinovich, M. Gensch, S. Kovalev, K.-J. Tielrooij. Gratinggraphe ne metamaterial as a platform for terahertz nonlinear photonics. ACS Nano 15 (1), 1145 (2021).
https://doi.org/10.1021/acsnano.0c08106
O.O. Vakhnenko, V.O. Vakhnenko. Development and analysis of novel integrable nonlinear dynamical systems on quasi-one-dimensional lattices. Two-component nonlinear system with the on-site and spatially distributed inertial mass parameters. Ukr. J. Phys. 69 (3), 168 (2024).
https://doi.org/10.15407/ujpe69.3.168
O.O. Vakhnenko. Semidiscrete integrable nonlinear systems generated by the new fourth-order spectral operator. Local conservation laws. J. Nonlin. Math. Phys. 18 (3), 401 (2011).
https://doi.org/10.1142/S1402925111001672
O.O. Vakhnenko. Four-wave semidiscrete nonlinear integrable system with PT -symmetry. J. Nonlin. Math. Phys. 20 (4), 606 (2013).
https://doi.org/10.1080/14029251.2013.865827
O.O. Vakhnenko. Four-component integrable systems inspired by the Toda and the Davydov-Kyslukha models, Wave Motion 88, 1 (2019).
https://doi.org/10.1016/j.wavemoti.2019.01.013
W.-X. Ma. A generating scheme for conservation laws of discrete zero curvature equations and its application. Comput. Math. Appl. 78 (10), 3422 (2019).
https://doi.org/10.1016/j.camwa.2019.05.012
O.O. Vakhnenko. Symmetry-broken canonizations of the semi-discrete integrable nonlinear Schr¨odinger system with background-controlled intersite coupling. J. Math. Phys. 57 (11), 113504 (2016).
https://doi.org/10.1063/1.4968244
O.O. Vakhnenko. Semi-discrete integrable nonlinear Schr¨odinger system with background-controlled inter-site resonant coupling. J. Nonlin. Math. Phys. 24 (2), 250 (2017).
https://doi.org/10.1080/14029251.2017.1316011
L.D. Faddeev, L.A. Takhtajan. Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987).
https://doi.org/10.1007/978-3-540-69969-9
G.-Z. Tu. A trace identity and its applications to the theory of discrete integrable systems. J. Phys. A: Math. Gen. 23 (17), 3903 (1990).
https://doi.org/10.1088/0305-4470/23/17/020
O.O. Vakhnenko, A.P. Verchenko. Nonlinear system of PT -symmetric excitations and Toda vibrations integrable by the Darboux-B¨acklund dressing method. Proc. R. Soc. A 477 (2256), 20210562 (2021).
https://doi.org/10.1098/rspa.2021.0562
O.O. Vakhnenko. Coupling-managed criticality in nonlinear dynamics of an integrable exciton-phonon system on a one-dimensional lattice. Low Temp. Phys. 47 (12), 1084 (2021).
https://doi.org/10.1063/10.0007084
O.O. Vakhnenko. Nonlinear dynamics of an integrable gauge-coupled exciton-phonon system on a regular one-dimensional lattice. Low Temp. Phys. 48 (3), 239 (2022).
https://doi.org/10.1063/10.0009543
O.O. Vakhnenko, A.P. Verchenko. Dipole-monopole alternative in nonlinear dynamics of an integrable gaugecoupled exciton-phonon system on a one-dimensional lattice. Eur. Phys. J. Plus 137 (10), 1176 (2022).
https://doi.org/10.1140/epjp/s13360-022-03335-w
O.O. Vakhnenko, V.O. Vakhnenko, A.P. Verchenko. Dipole-monopole alternative as the precursor of pseudoexcitonic chargeless half-mode in an integrable nonlinear exciton-phonon system on a regular one-dimensional lattice. Chaos, Solitons and Fractals 170, 113306 (2023).
https://doi.org/10.1016/j.chaos.2023.113306
O.O. Vakhnenko. Dipole-monopole crossover and chargeless half-mode in an integrable exciton-phonon nonlinear dynamical system on a regular one-dimensional lattice. Ukr. J. Phys. 68 (02), 108 (2023).
https://doi.org/10.15407/ujpe68.2.108
O.O. Vakhnenko, V.O. Vakhnenko. Dipole-monopole criticality and chargeless half mode in an integrable gaugecoupled pseudoexciton-phonon system on a regular onedimensional lattice. Phys. Rev. E 108 (02), 024223 (2023).
https://doi.org/10.1103/PhysRevE.108.024223
A.S. Davydov. Theory of Molecular Excitons (Plenum Press, 1971).
https://doi.org/10.1007/978-1-4899-5169-4
O.O. Vakhnenko, M.J. Velgakis. Multimode soliton dynamics in pertrubed ladder lattices. Phys. Rev. E 63 (1), 016612 (2001).
https://doi.org/10.1103/PhysRevE.63.016612
O.O. Vakhnenko. Enigma of probability amplitudes in Hamiltonian formulation of integrable semidiscrete nonlinear Schr¨odinger systems. Phys. Rev. E 77 (2), 026604 (2008).
https://doi.org/10.1103/PhysRevE.77.026604
A. Dewisme, S. Bouquet. First integrals and symmetries of time-dependent Hamiltonian systems. J. Math. Phys. 34 (3), 997 (1993).
https://doi.org/10.1063/1.530206
J¨u. Struckmeier, C. Riedel. Invariants for time-dependent Hamiltonian systems. Phys. Rev. E 64 (2), 026503 (2001).
Downloads
Published
How to Cite
Issue
Section
License
Copyright Agreement
License to Publish the Paper
Kyiv, Ukraine
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4. Duration.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5. Loyalty.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.
.png){kind=small}
