Gauge transformations and generating operators for the discrete Zakharov-Shabat system

Abstract

A gauge-covariant approach to the difference evolution equations (DEE), related to the discrete Zakharov-Shabat system ln(z), is proposed. The authors prove that the inverse scattering method, both for ln(z) and its gauge equivalent ln(z), may be treated as a generalised Fourier transform, thus showing that this interpretation is gauge independent. The proof of these facts is based on the expansions over certain complete sets of 'squared' solutions of ln(z) and ln(z), which in fact are spectral decompositions for the corresponding generating operators Lambda and Lambda . They calculate the operator Lambda solely in terms of Sn, the potential of ln(z). This obtains the following results (known already for the Zakharov-Shabat system): (i) the class of DEE, solvable with ln(z); (ii) their conserved quantities; and (iii) their hierarchies of Hamiltonian structures. The interrelation between the two hierarchies of Hamiltonian structures is obtained. The authors briefly discuss some interesting examples of gauge-equivalent DEE and the limit to the continuous case.