Nonlinear dynamics of quadratically cubic systems

O.V. Rudenko^{a, b, c, d, e}
^aLomonosov Moscow State University, Faculty of Physics, Leninskie Gory 1 build. 2, Moscow, 119991, Russian Federation
^bProkhorov General Physics Institute of the Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119991, Russian Federation
^cSchmidt Institute of Physics of the Earth of the Russian Academy of Sciences, Bolshaya Gruzinskaya ul. 10, Moscow, 123995, Russian Federation
^dN.I. Lobachevskii Nizhnii Novgorod State University, prosp. Gagarina 23, Nizhnii Novgorod, Russian Federation
^eBlekinge Institute of Technology, Karlskrona, Sweden

A modified form of the known nonlinear dynamics equations is proposed which uses quadratic relations to model cubic nonlinearity. It is shown that such quadratically cubic equations are sometimes amenable to exact solutions and that sometimes they make the original problem easier to analyze qualitatively. Occasionally, exact solutions provide a useful tool for studying new phenomena. Examples considered include nonlinear ordinary differential equations and nonlinear partial differential equations of the Hopf, Burgers, Korteveg—de Vries, and Schrödinger types. Some problems are solved exactly in the spatiotemporal and spectral representations. The unsolved problems potentially amenable to the proposed approach are listed.

PACS: 02.30.Jr, 05.45.−a, 42.65.−k, 43.25.+y (all)
DOI: 10.3367/UFNe.0183.201307b.0719
URL: https://ufn.ru/en/articles/2013/7/b/
Web of Science

000325716100002
Scopus

2-s2.0-84885986245
ADS NASA

2013PhyU...56..683R
Citation: Rudenko O V "Nonlinear dynamics of quadratically cubic systems" Phys. Usp. 56 683–690 (2013)

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Received: 9th, April 2013, revised: 29th, April 2013, accepted: 9th, April 2013

Оригинал: Руденко О В «Нелинейная динамика квадратично кубичных систем» УФН 183 719–726 (2013); DOI: 10.3367/UFNr.0183.201307b.0719

References (24) Cited by (19) Similar articles (20) ↓

V.F. Kovalev, D.V. Shirkov “Renormalization-group symmetries for solutions of nonlinear boundary value problems” Phys. Usp. 51 815–830 (2008)
G.I. Broman, O.V. Rudenko “Submerged Landau jet: exact solutions, their meaning and application” Phys. Usp. 53 91–98 (2010)
E.P. Zemskov “Turing patterns and Newell—Whitehead—Segel amplitude equation” Phys. Usp. 57 1035–1037 (2014)
S.N. Gordienko “Irreversibility and the probabilistic treatment of the dynamics of classical particles” Phys. Usp. 42 573–590 (1999)
P.S. Landa, D.I. Trubetskov, V.A. Gusev “Delusions versus reality in some physics problems: theory and experiment” Phys. Usp. 52 235–255 (2009)
E.N. Rumanov “Critical phenomena far from equilibrium” Phys. Usp. 56 93–102 (2013)
A. Loskutov “Dynamical chaos: systems of classical mechanics” Phys. Usp. 50 939–964 (2007)
A.V. Kukushkin “A technique for solving the wave equation and prospects for physical applications arising therefrom” Phys. Usp. 36 (2) 81–93 (1993)
A.V. Borisov, A.O. Kazakov, S.P. Kuznetsov “Nonlinear dynamics of the rattleback: a nonholonomic model” Phys. Usp. 57 453–460 (2014)
A.V. Borisov, I.S. Mamaev “Strange attractors in rattleback dynamics” Phys. Usp. 46 393–403 (2003)
A.V. Belinsky, M.Kh. Shulman “Quantum nature of a nonlinear beam splitter” Phys. Usp. 57 1022–1034 (2014)
V.I. Klyatskin “Statistical topography and Lyapunov exponents in stochastic dynamical systems” Phys. Usp. 51 395–407 (2008)
V.P. Silin, P.V. Silin “Bifurcation properties of the bremsstrahlung harmonics generated by a pumping field in plasmas” Phys. Usp. 50 729–740 (2007)
B.Ya. Zel’dovich, M.J. Soileau “Bi-frequency pendulum on a rotary platform: modeling various optical phenomena” Phys. Usp. 47 1239–1255 (2004)
P.S. Landa, Ya.B. Duboshinskii “Self-oscillatory systems with high-frequency energy sources” Sov. Phys. Usp. 32 723–731 (1989)
V.S. Zapasskii “On electromagnetically induced transparency in the degenerate Λ-scheme” Phys. Usp. 52 179–181 (2009)
A.N. Pavlov, V.S. Anishchenko “Multifractal analysis of complex signals” Phys. Usp. 50 819–834 (2007)
V.V. Brazhkin “Why does statistical mechanics 'work' in condensed matter?” Phys. Usp. 64 1049–1057 (2021)
S.V. Sazonov “Superluminal electromagnetic solitons in nonequilibrium media” Phys. Usp. 44 631–644 (2001)
A.N. Oraevskii “Superluminal waves in amplifying media” Phys. Usp. 41 1199–1209 (1998)

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