From generalized Fourier transforms to spectral curves for the Manakov hierarchy. I. Generalized Fourier transforms

The generalized Fourier transforms (GFTs) for the hierarchies of multi-component models of Manakov type are revisited. Our aim is to adapt GFT so that one could treat the whole hierarchy of nonlinear evolution equations simultaneously. To this end we consider the potential Q of the Lax operator L as local coordinate on some symmetric space depending on an infinite number of variables t, and $$z_k$$ , $$k=1, 2, \ldots $$ . The dependence on $$z_k$$ is determined by the k-th higher flow (conserved density) of the hierarchy. Thus we have an infinite set of commuting operators with common fundamental analytic solution. We analyze the properties of the resolvent and thus determine the spectral properties of L. Then we derive the generalized Fourier transforms that linearize this hierarchy of NLEE and establish their fundamental properties as well as dynamical compatibility of each two pairs of such flows. Using the classical R-matrix approach we derive the Poisson brackets between all conserved quantities first assuming that Q is a quasi-periodic function of t. Next taking the limit when the period tends to $$\infty $$ , we derive the Poisson brackets between the scattering data of L. In addition we analyze a possible relation of our approach to the one based on the $$\tau $$ -function that could lead to multi-dimensional integrable equations.