Christopher W. Curtis

Vortex dynamics in nonlinear free surface flows

The two-dimensional motion of point vortices in an inviscid fluid with a free surface and an impenetrable bed is investigated. The work is based on forming a closed system of equations for surface variables and vortex positions using a variant of the Ablowitz, Fokas, and Musslimani formulation [M. J. Ablowitz, A. S. Fokas, and Z. H. Musslimani, J. Fluid Mech. 562, 313–343 (2006)] of the water-wave free-surface problem. The equations are approximated with a dealiased spectral method making use of a high-order approximation of the Dirichlet-Neumann operator and a high-order time-stepping scheme. Numerical simulations reveal that the combination of vortex motion and solid bottom boundary yields interesting dynamics not seen in the case of vortex motion in an infinitely deep fluid. In particular, strong deformations of the free surface, including non-symmetric surface profiles and regions of large energy concentration, are observed. Our simulations also uncover a rich variety of vortex trajectories including ...

Conservation laws and non-decaying solutions for the Benney–Luke equation

A long wave multi-dimensional approximation of shallow-water waves is the bi-directional Benney–Luke (BL) equation. It yields the well-known Kadomtsev–Petviashvili (KP) equation in a quasi one-directional limit. A direct perturbation method is developed; it uses underlying conservation laws to determine the slow evolution of parameters of two space-dimensional, non-decaying solutions to the BL equation. These non-decaying solutions are perturbations of recently studied web solutions of the KP equation. New numerical simulations, based on windowing methods which are effective for non-decaying data, are presented. These simulations support the analytical results and elucidate the relationship between the KP and the BL equations and are also used to obtain amplitude information regarding particular web solutions. Additional dissipative perturbations to the BL equation are also studied.

On the evolution of perturbations to solutions of the Kadomtsev-Petviashvilli equation using the Benney-Luke equation

The Benney–Luke equation, which arises as a long wave asymptotic approximation of water waves, contains the Kadomtsev–Petviashvilli (KP) equation as a leading-order maximal balanced approximation. The question analyzed is how the Benney–Luke equation modifies the so-called web solutions of the KP equation. It is found that the Benney–Luke equation introduces dispersive radiation which breaks each of the symmetric soliton-like humps well away from the interaction region of the KP web solution into a tail of multi-peaked oscillating profiles behind the main solitary hump. Computation indicates that the wave structure is modified near the center of the interaction region. Both analytical and numerical techniques are employed for working with non-periodic, non-decaying solutions on unbounded domains.

On the existence of real spectra in

This paper investigates the addition of complex potentials to Schr?dinger equations with honeycomb lattice potentials. While self-adjointness is lost, symmetry with respect to the combined action of parity and time-reversal is maintained. Necessary and sufficient conditions are found which establish whether the spectra of the perturbed Schr?dinger equation remains real or enters the complex plane after small perturbations. Numerical simulations show the impact of larger perturbations. In particular, nonlocal near degeneracies between dispersion bands are discussed.