Two-photon propagation of light and the modified Liouville equation

A. M. Kamchatnov^ab, M. V. Pavlov<sup>c

a</sup> Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow
^b Moscow Institute of Physics and Technology (National Research University), Dolgoprudny, Moscow Oblast, Russia
^c Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia

Abstract: We show that the system of nonlinear equations of two-photon propagation of light with real amplitudes of the envelopes can be solved in general form by the classical Liouville method. This system, like other similar systems of Darboux-integrable equations, is related to the modified Liouville equation, and the found solution also provides general solutions of such modified equations. We conclude that the Liouville method provides an effective way to integrate a class of concrete system that admit Darboux integration.

Keywords: Darboux integration, generalized Liouville equation, two-photon propagation of light.

PACS: 02.30.Ik, 05.45.Yv, 42.65.Tg

MSC: 37K10

Received: 13.02.2020
Revised: 19.03.2020

DOI: 10.4213/tmf9887

 English version: Theoretical and Mathematical Physics, 2020, 204:2, 1093–1099

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