Two-photon propagation of light and the modified Liouville equation
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TWO-PHOTON PROPAGATION OF LIGHT AND THE MODIFIED LIOUVILLE EQUATION A. M. Kamchatnov∗ and M. V. Pavlov†
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
DOI: 10.1134/S0040577920080097
1. Introduction One of the main methods for studying the propagation of nonlinear waves is the inverse scattering method, which is applicable to a rather wide class of equations with important physical applications. In particular, this method allows finding multisoliton or multiphase quasiperiodic solutions of such equations and studying their perturbations or modulations (see, e.g., [1], [2]). But for some problems, soliton propagation is not typical, and a more general form of the solution that allows expressing the result of wave evolution in terms of the initial data is needed. A typical example of this kind of problem is the classical problem of second harmonic generation, where there is a light pulse of the same frequency at the entrance to the medium and it is necessary to know the shape of the pulse with a doubled frequency at the exit from the medium (see, e.g., [3], [4]). If we let E1 and E2 denote the complex envelopes of two harmonic fields, then we can write the corresponding equations for the propagation of the interacting waves as ∂χ E1 = 2E2 E1∗ ,
∂τ E2 = E12 ,
(1)
where χ and τ are the characteristic coordinates for the linear propagation of the waves χ=
v1 v2 x t− , v1 − v2 v2
τ=
v1 v2 x t− v1 − v2 v1
(2)
∗
Institute of Spectroscopy, RAS, Moscow, Troisk, Russia; Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Oblast, Russia, e-mail: [email protected], [email protected]. †
Lebedev Physical Institute, RAS, Moscow, Russia.
The research of M. V. Pavlov was supported by the Russian Foundation for Basic Research (Grant No. 20-0100157 A). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 297–304, August, 2020. Received February 13, 2020. Revised March 19, 2020. Accepted March 30, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-1093
1093
and v1 and v2 are corresponding group velocities. As noted in [5], if the envelopes E1 and E2 are real functions, then they remain real in the process of propagation, and eliminating E2 leads to the equation for E1 , sχτ = es , s = log(4E12 ). (3) This is the famous Liouville equation [6], which admits a general solution in a closed form expressed in terms of two arbitrary functions. This allows a rather complete study [7] of the process of generating
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