Dispersionful analogue of the Whitham hierarchy

The dispersionful analogue, by means of Lax formalism, of the zero-genus universal Whitham hierarchy together with its algebraic orbit finite-field reductions is considered. The theory is illustrated by several significant examples.

💡 Research Summary

The paper presents a systematic construction of a dispersionful analogue of the zero‑genus universal Whitham hierarchy using the Lax formalism. Historically, Whitham hierarchies have been employed to describe slow modulations of nonlinear wave fields, but most developments have focused on the dispersionless limit, where the underlying equations reduce to quasi‑classical Hamilton–Jacobi type systems. The authors aim to extend this framework by incorporating explicit dispersion effects, thereby bridging the gap between classical Whitham theory and fully dispersive integrable hierarchies such as KP, Toda, and KdV.

The construction begins with the standard algebraic description of the zero‑genus Whitham hierarchy: a Riemann surface (in this case a rational curve) equipped with a set of local coordinates and a generating differential. From this geometric data one derives a Poisson structure and a set of Hamiltonians that generate the hierarchy’s flows. To introduce dispersion, the authors replace the classical generating function by a pair of pseudo‑differential operators (L) (the Lax operator) and (M) (the Orlov–Schulman operator) satisfying the canonical commutation relation (