Coupled KdV equations with multi-Hamiltonian structures

Abstract

We present a large class of systems of N-equations which possess ( N+1) purely differential, compatible Hamiltonian structures. Our generic equations are isospectral to [( Σ N−1 0 ε i λ 2 i ) ∂ 2+ Σ N−1 0 υ i λ 2 i ] ψ= λ 2 N ψ, but our class also includes degenerate, nondispersive systems which are unrelated to this linear problem. Embedded in this class, for each N, there are N distinct coupled KdV systems. When N = 2 our class includes 3 known equations: dispersive water waves, Ito's equation and reduced Benney's equations. These equations are thus tri-Hamiltonian. We also present 2 examples of 3-component quadri-Hamiltonian systems, which generalise the above mentioned dispersive and nondispersive water waves.