The Infinite-genus Limit of the Whitham Equations

As is well known [1], [2], the modulation (Whitham) equations for integrable systems describe gradual variations of spectral Riemann surfaces, whose local structure defines x-, t-oscillating finite-gap potentials. The wave-dynamics problems leading to the modulation equations are typically connected with studying the long-time evolution of large-scale initial data [3], [4] and involve the semiclassical asymptotics of the inverse scattering transform. In this asymptotics, the typical scale of modulations 1/ is necessarily proportional to a global number of degrees of freedom g 1 (here we assume no small parameters in the original equation; instead, we consider large-scale ∼ 1/ initial data). Then, asymptotically, the solution manifests itself as a modulated N -gap potential, where N g [5], [6]. The whole point of the modulation theory is thus to reduce a complicated system with many degrees of freedom to a simpler one with a few degrees of freedom. But the modulation equations can be considered on their own outside any relation to the semiclassical initial value problem. In this connection, the question of the possibility of the infinite-genus limit for the modulation equations can naturally arise. It is clear that to provide the existence of such a limit, some special way of the limiting transition for the spectral curve should be indicated. In this paper, we consider the limit as N → ∞ for the N -phased averaged Korteweg–de Vries (KdV) equation. The KdV equation is taken in the form