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Hamiltonian formulation for solitary waves propagating on a variable background. (English) Zbl 0934.76011

Summary: 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.

MSC:

76B25 Solitary waves for incompressible inviscid fluids
35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
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