System of equations for stimulated combination scattering and the related double periodic $A_n^{(1)}$ Toda chains

V. A. Andreev

P. N. Lebedev Physical Institute, Russian Academy of Sciences

Abstract: We consider a system of equations describing stimulated combination scattering of light. We show that solutions of this system are expressed in terms of two solutions of the sine-Gordon equation that are related to each other by a Bäcklund transformation. We also show that this system is integrable and admits a Zakharov–Shabat pair. In the general case, the system of equations for the Bäcklund transformation of periodic $A_n^{(1)}$ Toda chains is also shown to be integrable and to have a Zakharov–Shabat pair.

Keywords: combination scattering, Toda chain, Bäcklund transformation, Zakharov–Shabat pair.

Received: 29.03.2007
Revised: 22.06.2007

DOI: 10.4213/tmf6230

 English version: Theoretical and Mathematical Physics, 2008, 156:1, 1020–1027

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