Stationary coupled KdV systems and their Stäckel representations (Q6590518)

In this paper, the authors explore the stationary coupled Korteweg-de Vries (cKdV) systems, demonstrating that every \(N\)-field stationary cKdV system can be expressed, after a careful reparameterization of jet variables, as a classical separable Stäckel system in \(N+1\) different forms. These different parameterizations are mapped explicitly to separation variables. The study reveals that each pair of Stäckel representations of the same system, when extended into phase space via Casimir variables, is connected by a finite-dimensional Miura map. This connection leads to an \((N+1)\)-Hamiltonian formulation of the stationary cKdV system.\N\NThe paper begins by reviewing the classical construction of the \(N\)-component cKdV hierarchy derived from the energy-dependent Schrödinger spectral problem. The system of equations is given by the compatibility condition for the Schrödinger equation:\N\[\N\psi_{xx} + Q\psi = 0, \quad \psi_{t_k} = \frac{1}{2} P_k \psi_x - \frac{1}{4} (P_k)_x \psi, \quad k = 1, 2, \dots\N\]\Nwhere \(Q\) is the sum over dynamical fields \(u_i\), and \(P_k\) are functions of the spectral parameter and jet variables. These form a hierarchy of evolution equations that lead to the construction of a multi-Hamiltonian system. The investigation focuses on the structure of the stationary cKdV system, which arises by restricting the cKdV hierarchy to stationary manifolds, \(\mathcal{M}_n\). The authors show that Hamiltonian foliations of the stationary manifold \(\mathcal{M}_n\) provide a natural way to derive separation (spectral) curves of Stäckel systems.\N\NThe main result of the paper is the proof that each stationary cKdV system can be represented as a Stäckel system on \(\mathcal{M}_n\) in \(N+1\) different ways. Moreover, these representations are connected by Miura maps, generating an \((N+1)\)-Hamiltonian structure. This provides a comprehensive understanding of the relationship between Stäckel systems and the stationary cKdV systems, offering a new approach to multi-Hamiltonian formulations in integrable systems.\N\NThe paper concludes by discussing examples, including the dispersive water waves (DWW) hierarchy for various values of \(N\), \(n\), and \(m\), and compares the results with previous work on related systems.