Nonlinear dynamics of quadratically cubic systems
a Lomonosov Moscow State University, Department of Physics, Leninskie Gory 1 build. 2, Moscow, 119991, Russian Federation
b Prokhorov General Physics Institute of the Russian Academy of Sciences, ul. Vavilova 38, Moscow, 119942, Russian Federation
c Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences, Bolshaya Gruzinskaya ul. 10, Moscow, 123995, Russian Federation
d N.I. Lobachevskii Nizhnii Novgorod State University, prosp. Gagarina 23, Nizhnii Novgorod, Russian Federation
e Blekinge 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.