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


  • O.O. Vakhnenko Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine
  • V.O. Vakhnenko Department of Dynamics of Deformable Solids, Subbotin Institute of Geophysics, Nat. Acad. of Sci. of Ukraine



nonlinear dynamics, integrable system, quasi-one-dimensional lattice, stable displacements, unstable displacements


The main principles of developing the evolutionary nonlinear integrable systems on quasi-onedimensional lattices are formulated in clear mathematical and physical terms discarding the whimsical mathematical formulations and computer-addicted presentations. These basic principles are substantiated by the actual development of novel semi-discrete integrable nonlinear system, whose auxiliary spectral and evolutionary operators are given by 4 × 4 square matrices. The procedure of reduction from the prototype nonlinear integrable system with twelve field functions to the physically meaningful nonlinear integrable system with four field functions is described in details prompted by our previous cumulative experience. The obtained ultimate semi-discrete nonlinear integrable system comprises the two subsystems of essentially distinct physical origins. Thus, the first subsystem is the subsystem of the Toda type. It is characterized by the on-site (spatially local) mass parameter and the positively defined elasticity coefficient. In contrast, the second subsystem is characterized by the spatially distributed mass parameters and the negatively defined elasticity coefficient responsible for the low-amplitude instability. We believe our scrupulous consideration of all main steps in developing the semidiscrete nonlinear integrable systems will be useful for the researchers unfamiliar with the numerous stumbling blocks inevitable in such an interesting and prospective scientific field as the theory of semi-discrete nonlinear integrable systems.


N.M. Krylov, N.N. Bogoliubov. Introduction to Non-Linear Mechanics (Princeton University Press, 1950) [ISBN: 9780691079851].

N.N. Bogoliubov, Y.A. Mitropolsky. Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon and Breach Science Publishers, 1961) [ISBN-10: 067720051X].

Yu.A. Mitropolskii, Nguen Van Dao. Applied Asymptotic Methods in Nonlinear Oscillations (Springer Science + Business Media, 1997).

A.R.E. Oliveira. History of Krylov-Bogoliubov-Mitropolsky methods of nonlinear oscillations. Adv. Histor. Stud. 6, 40 (2017).

E. Fermi, P. Pasta, S. Ulam, M. Tsingou. Studies of the nonlinear problems. I, Los Alamos Report LA-1940, 1 (1955).

M. Toda. Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan 22 (2), 431 (1967).

M. Toda. Wave propagation in anharmonic lattices. J. Phys. Soc. Japan 23 (3), 501 (1967).

M. Toda. Studies of a non-linear lattice. Phys. Rep. 18 (1), 1 (1975).

M.J. Ablowitz. Lectures on the inverse scattering transform. Stud. Appl. Math. 58 (1), 17 (1978).

L.D. Faddeev, L.A. Takhtajan. Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, 1987).

T. Tsuchida, H. Ujino, M. Wadati. Integrable semidiscretization of the coupled nonlinear Schr¨odinger equations. J. Phys. A: Math. Gen. 32 (11), 2239 (1999).

O.O. Vakhnenko. Nonlinear beating excitations on ladder lattice. J. Phys. A: Math. Gen. 32 (30), 5735 (1999).

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).

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

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).

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).

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).

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).

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).

O.O. Vakhnenko, M.J. Velgakis. Multimode soliton dynamics in pertrubed ladder lattices. Phys. Rev. E 63 (1), 016612 (2001).

O.O. Vakhnenko. Enigma of probability amplitudes in Hamiltonian formulation of integrable semidiscrete nonlinear Schr¨odinger systems. Phys. Rev. E 77(2), 026604 (2008).

S. Timoshenko. Vibration Problems in Engineering (D. Van Nostrand Company, Inc., 1937).

V.F. Nesterenko. Dynamics of Heterogeneous Materials (Springer-Verlag, 2001).

Yu.B. Gaididei, C. Gorria, R. Berkemer, A. Kawamoto, T. Shiga, P.L. Christiansen, M.P. Sørensen, J. Starke. Controlling traffic jams by time modulating the safety distance. Phys. Rev. E 88 (4), 042803 (2013).

Yu.B. Gaididei, P.L. Christiansen, M.P. Sørensen, J.J. Rasmussen. Analytical solutions of pattern formation for a class of discrete Aw-Rascle-Zhang traffic models. Commun. Nonlin. Sci. Numer. Simul. 73, 391 (2019).

Yu.B. Gaididei, C. Marschler, M.P. Sørensen, P.L. Christiansen, J.J. Rasmussen. Pattern formation in flows of asymmetrically interacting particles: Peristaltic pedestrian dynamics as a case study. Evolution Equations and Control Theory 8 (1), 73 (2019).




How to Cite

Vakhnenko, O., & Vakhnenko, V. (2024). 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. Ukrainian Journal of Physics, 69(3), 168.



General physics