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