Non-interacting gravity waves on the surface of a deep fluid

We study the interaction of gravity waves on the surface of an infinitely deep ideal fluid. Starting from Zakharov's variational formulation for water waves we derive an expansion of the Hamiltonian to an arbitrary order, in a manner that avoids a laborious series reversion associated with expressing the velocity potential in terms of its value at the free surface. The expansion kernels are shown to satisfy a recursion relation enabling us to draw some conclusions about higher-order wave-wave interaction amplitudes, without referring to the explicit forms of the individual lower-order kernels. In particular, we show that unidirectional waves propagating in a two-dimensional flow do not interact nonlinearly provided they fulfill the energy-momentum conservation law. Switching from the physical variables to the so-called normal variables we explain the vanishing of the amplitudes of fourth- and certain fifth-order non-generic resonant interactions reported earlier and outline a procedure for finding the one-dimensional wave vector configurations for which the higher order interaction amplitudes become zero on the resonant hypersurfaces.Comment: 13 page

Isospectral Potentials from Modified Factorization

Factorization of quantum mechanical potentials has a long history extending back to the earliest days of the subject. In the present paper, the non-uniqueness of the factorization is exploited to derive new isospectral non-singular potentials. Many one-parameter families of potentials can be generated from known potentials using a factorization that involves superpotentials defined in terms of excited states of a potential. For these cases an operator representation is available. If ladder operators are known for the original potential, then a straightforward procedure exists for defining such operators for its isospectral partners. The generality of the method is illustrated with a number of examples which may have many possible applications in atomic and molecular physics.Comment: 8 pages, 4 figure