4 research outputs found
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