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The Gurevich-Zybin system. (English) Zbl 1068.37062

Summary: We present three different linearizable extensions of the Gurevich-Zybin system. Their general solutions are found by reciprocal transformations. We rewrite the Gurevich-Zybin system as a Monge-Ampère equation. By application of reciprocal transformation, this equation is linearized. Infinitely many local Hamiltonian structures, local Lagrangian representations, local conservation laws and local commuting flows are found. Moreover, all commuting flows can be written as Monge-Ampère equations similar to the Gurevich-Zybin system. The Gurevich-Zybin system describes the formation of large scale structure in the Universe.
Second harmonic wave generation is known in nonlinear optics. We prove that the Gurevich-Zybin system is equivalent to a degenerate case of second harmonic generation. Thus, the Gurevich-Zybin system is recognized as a degenerate first negative flow of two-component Harry-Dym hierarchy up to two Miura-type transformations. A reciprocal transformation between the Gurevich-Zybin system and degenerate case of the second harmonic generation system is found. A new solution for second harmonic generation is presented in implicit form.

MSC:

37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
83B05 Observational and experimental questions in relativity and gravitational theory
83C35 Gravitational waves
78A60 Lasers, masers, optical bistability, nonlinear optics
35J60 Nonlinear elliptic equations
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