Abstract
Nonlinear differential-difference equations (nonlinear lattices) exhibit single soliton solutions in the form of 1-breathers. These solutions are time-periodic and exponentially localized in space. Necessary condition for their existence is the upper bounds on the linear spectrum of small perturbations around stationary point of the system. Unlike continuous models, integrability property do not help nonlinear lattices to have multi-breather solutions. The C-integrable Liouville lattice is discussed in view to get moving breathers, i.e. true time-periodic multi-solitons with free velocity and amplitude. We construct analogs of instanton solutions which are localized both in space and time.
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REFERENCES
Theory of Solitons: The Inverse Scattering Method, Ed. by S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov (Nauka, Moscow, 1980; Springer, New York, 1984).
S. Flach, ‘‘Existence and properties of discrete breathers,’’ in Nonlinear Physics Theory and Experiment, Ed. by E. Alfinito (World Scientific, Singapore, 1996), pp. 390–397.
S. Takeno, ‘‘Theory of stationary anharmonic localized modes in solids,’’ J. Phys. Soc. Jpn. 61, 2821–2834 (1992).
S. Flasch and S. R. Willis, ‘‘Discrete breathers,’’ Phys. Rep. 298, 181–294 (1998).
D. Hennig and G. P. Tsironis, ‘‘Wave transition in nonlinear lattices,’’ Phys. Rep. 310, 333–432 (1999).
V. E. Adler and S. Ya. Startsev, ‘‘On discrete analogues of Liouville equation,’’ Theor. Math. Phys. 121, 271–284 (1999).
J. Liouville, ‘‘Sur l’equation aux differences partielles \(\frac{d^{2}\log\lambda}{dudv}\pm\frac{\lambda}{2a^{2}}=0\),’’ J. Math. Pures Appl. 18, 71–72 (1853).
B. Birnir, ‘‘Qualitative analysis of radiating breathers,’’ Comm. Pure Appl. Math. 47, 103–117 (1994).
H. Segur and M. D. Kruskal, ‘‘Nonexistence of small-amplitude breather solutions in \(phi^{4}\) theory,’’ Phys. Rev. Lett. 58, 747–751 (1987).
W. Magnus and S. Winkler, Hill’s Equation (Interscience, New York, 1966).
F. Colombini, V. Petkov, and J. Rauch, ‘‘Exponential growth for the wave equation with compact time-periodic positive potential,’’ Comm. Pure Appl. Math. 62, 565–582 (2009).
F. Colombini and J. Rauch, ‘‘Smooth localized parametric resonance for wave equations,’’ J. Reine Angew. Math. 616, 1–14 (2008).
V. M. Eleonskii, N. E. Kulagin, N. S. Novozhilova, and V. P. Silin, ‘‘Asymptotic expansions and qualitative analysis of finite-dimensional models in nonlinear field theory,’’ Teor. Mat. Fiz. 60, 395–403 (1984).
M. Toda, ‘‘Vibration of a chain with nonlinear interaction,’’ J. Phys. Soc. Jpn. 22, 431–436 (1967).
M. Toda, ‘‘Wave propagation in anharmonic lattices,’’ J. Phys. Soc. Jpn. 23, 501–506 (1967).
V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
S. Aubry and T. Cretegny, ‘‘Mobility and reactivity of discrete breathers,’’ Phys. D (Amsterdam, Neth.) 119, 34–46 (1998).
G. Darboux, Lecons sur la theorie generale des surfaces et les applications geometriques du calcul infinitesimal (Gauthier-Villars, Paris, 1896).
V. V. Sokolov and A. V. Zhiber, ‘‘On the Darboux integrable hyperbolic equations,’’ Phys. Lett. A 208, 303–308 (1995).
J. L. Marin, J. C. Eilbeck, and F. M. Russell, ‘‘2-D breathers and applications,’’ Lect. Notes Phys. 542, 293–305 (2000).
Y. Yang and Y. Zhu, ‘‘Darboux–Bäcklund transformation, breather and rogue wave solutions for Ablowitz–Ladik equation,’’ Optik 217, 164920 (2020).
A. K. Pogrebkov, ‘‘Singular solitons: An example of a sinh-Gordon equation,’’ Lett. Math. Phys. 5, 277–285 (1981).
Instantons in Gauge Theories, Ed. by M. A. Shifman (World Scientific, Singapore, 1994).
H. J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Scientific, Singapore, 2012).
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Novokshenov, V.Y. Localization in the Liouville Lattice and Movable Discrete Breathers. Lobachevskii J Math 42, 1210–1218 (2021). https://doi.org/10.1134/S1995080221060214
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DOI: https://doi.org/10.1134/S1995080221060214