Lax pairs and Fourier analysis: The case of sine-Gordon and Klein-Gordon equations

摘要

In this paper we construct a new Lax pair for the Klein-Gordon equation. The structure algebra of this Lax pair is the algebra T A_2 of upper triangular Toeplitz block matrices with su(2) blocks. For the suitable choice of the values of the spectral parameter, the integrals of motion, obtained from the holonomy of the spatial part of the Lax pair, have simple expressions in terms of the Fourier data. We compare these integrals to the corresponding integrals of the sine-Gordon system.