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.
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Notes
We note that system (6), strictly speaking, yields the integral \(d^2+n^2=(\psi'(\chi))^2\) with an arbitrary function \(\psi(\chi)\). This case is trivially reduced to the particular case that we consider by an elementary change of the variable \(\chi\), \(\zeta=\psi(\chi)\).
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Funding
The research of M. V. Pavlov was supported by the Russian Foundation for Basic Research (Grant No. 20-01-00157 A).
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Kamchatnov, A.M., Pavlov, M.V. Two-photon propagation of light and the modified Liouville equation. Theor Math Phys 204, 1093–1099 (2020). https://doi.org/10.1134/S0040577920080097
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DOI: https://doi.org/10.1134/S0040577920080097