Hamiltonian formulation for solitary waves propagating on a variable background

Abstract

Solitary waves propagating on a variable background are conventionally described by the variable-coefficient Korteweg-de Vries equation. However, the underlying physical system is often Hamiltonian, with a conserved energy functional. Recent studies for water waves and interfacial waves have shown that an alternative approach to deriving an appropriate evolution equation, which asymptotically approximates the Hamiltonian, leads to an alternative variable-coefficient Korteweg-de Vries equation, which conserves the underlying Hamiltonian structure more explicitly. This paper examines the relationship between these two evolution equations, which are asymptotically equivalent, by first discussing the conservation laws for each equation, and then constructing asymptotically a slowly-varying solitary wave.