This article is cited in 24 papers

METHODOLOGICAL NOTES

Nonlinear dynamics of quadratically cubic systems

O. V. Rudenko<sup>abcde

a</sup> Lobachevsky State University of Nizhny Novgorod
^b Prokhorov General Physics Institute, Russian Academy of Sciences
^c Faculty of Physics, Lomonosov Moscow State University
^d Schmidt Institute of the Earth, Russian Academy of Scienses
^e Blekinge Institute of Technology

Abstract: We propose a modified form of the well-known nonlinear dynamic equations with quadratic relations used to model a cubic nonlinearity. We show that such quadratically cubic equations sometimes allow exact solutions and sometimes 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 Hopf, Burgers, Korteweg – de Vries, and nonlinear Schrödinger partial differential equations. Some problems are solved exactly in the space–time and spectral representations. Unsolved problems potentially solvable by the proposed approach are listed.

PACS: 02.30.Jr, 05.45.-a, 42.65.-k, 43.25.+y

Received: April 9, 2013
Revised: April 29, 2013
Accepted: April 9, 2013

DOI: 10.3367/UFNr.0183.201307b.0719

 English version: Physics–Uspekhi, 2013, 56:7, 683–690

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