Annalisa Calini

Homoclinic chaos increases the likelihood of rogue wave formation

We numerically investigate dispersive perturbations of the nonlinear Schrodinger (NLS) equation, which model waves in deep water. We observe that a chaotic regime greatly increases the likelihood of rogue wave formation. These large amplitude waves are well modeled by higher order homoclinic solutions of the NLS equation for which the spatial excitations have coalesced to produce a wave of maximal amplitude. Remarkably, a Melnikov analysis of the conditions for the onset of chaos identifies the observed maximal amplitude homoclinic solutions as the persistent hyperbolic structures throughout the perturbed dynamics.

BÄCKLUND TRANSFORMATIONS AND KNOTS OF CONSTANT TORSION

The Backlund transformation for pseudospherical surfaces, which is equivalent to that of the sine-Gordon equation, can be restricted to give a transformation on space curves that preserves constant torsion. We study its effects on closed curves (in particular, elastic rods) that generate multiphase solutions for the vortex filament flow (also known as the Localized Induction Equation). In doing so, we obtain analytic constant-torsion representatives for a large number of knot types.

Remarks on KdV-type Flows on Star-Shaped Curves

Abstract We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into R P 1 . We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution equation for closed star-shaped planar curves, discovered by Pinkall, has the Schwarzian KdV equation as its projectivization. (For both flows, the curvature evolves by the KdV equation.) Using algebro-geometric methods and the relation of group-based moving frames to AKNS-type representations, we construct examples of closed solutions of Pinkall’s flow associated with periodic finite-gap KdV potentials.

Finite-Gap Solutions of the Vortex Filament Equation: Genus One Solutions and Symmetric Solutions

AbstractFor the class of quasiperiodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction, and the geometry of the evolving curves. We give a complete description of genus one solutions, including geometrically interesting special cases such as Euler elastica, constant torsion curves, and self-intersecting filaments. We also prove generalizations of these connections to higher genus.

Dynamical criteria for rogue waves in nonlinear Schrödinger models

We investigate rogue waves in deep water in the framework of the nonlinear Schrodinger (NLS) and Dysthe equations. Amongst the homoclinic orbits of unstable NLS Stokes waves, we seek good candidates to model actual rogue waves. In this paper we propose two selection criteria: stability under perturbations of initial data, and persistence under perturbations of the NLS model. We find that requiring stability selects homoclinic orbits of maximal dimension. Persistence under (a particular) perturbation selects a homoclinic orbit of maximal dimension all of whose spatial modes are coalesced. These results suggest that more realistic sea states, described by JONSWAP power spectra, may be analyzed in terms of proximity to NLS homoclinic data. In fact, using the NLS spectral theory, we find that rogue wave events in random oceanic sea states are well predicted by proximity to homoclinic data of the NLS equation.

A note on a Bäcklund transformation for the continuous Heisenberg model

Abstract A formula for the Backlund transformation for the continuous Heisenberg model is given and immersed singular knots in three-space are constructed.

Stability of small-amplitude torus knot solutions of the localized induction approximation

We study the linear stability of small-amplitude torus knot solutions of the localized induction approximation equation for the motion of a thin vortex filament in an ideal fluid. Such solutions can be constructed analytically through the connection with the focusing nonlinear Schrodinger equation using the method of isoperiodic deformations. We show that these (p, q) torus knots are generically linearly unstable for p q, in contrast with an earlier linear stability study by Ricca (1993 Chaos 3 83–95; 1995 Chaos 5 346; 1995 Small-scale Structures in Three-dimensional Hydro and Magneto-dynamics Turbulence (Lecture Notes in Physics vol 462) (Berlin: Springer)). We also provide an interpretation of the original perturbative calculation in Ricca (1995), and an explanation of the numerical experiments performed by Ricca et al (1999 J. Fluid Mech. 391 29–44), in light of our results.

Connecting geometry, topology and spectra for finite-gap NLS potentials

Using the connection between closed solution curves of the vortex filament flow and quasiperiodic solutions of the nonlinear Schrodinger equation (NLS), we relate the knot types of finite-gap solutions to the Floquet spectra of the corresponding NLS potentials, in the special case of small amplitude curves close to multiply covered circles.

Rogue Waves in Higher Order Nonlinear Schrödinger Models

We discuss physical and statistical properties of rogue wave generation in deep water from the perspective of the focusing Nonlinear Schrodinger equation and some of its higher order generalizations. Numerical investigations and analytical arguments based on the inverse spectral theory of the underlying integrable model, perturbation analysis, and statistical methods provide a coherent picture of rogue waves associated with nonlinear focusing events. Homoclinic orbits of unstable solutions of the underlying integrable model are certainly candidates for extreme waves, however, for more realistic models such as the modified Dysthe equation two novel features emerge: (a) a chaotic sea state appears to be an important mechanism for both generation and increased likelihood of rogue waves; (b) the extreme waves intermittently emerging from the chaotic background can be correlated with the homoclinic orbits characterized by maximal coalescence of their spatial modes.

Squared eigenfunctions and linear stability properties of closed vortex filaments

We develop a general framework for studying the linear stability of closed solutions of the vortex filament equation (VFE), based on the correspondence between the VFE and the nonlinear Schrodinger (NLS) equation provided by the Hasimoto map, and on the construction of solutions of the linearized equations in terms of NLS squared eigenfunctions. In particular, we show that the differential of the Hasimoto map is a one-to-one correspondence between curve variations and perturbations of NLS potentials induced by squared eigenfunctions. We apply this framework to vortex filaments associated with periodic finite-gap NLS potentials in the genus one case, and for cnoidal potentials we characterize the stability of the associated filaments in terms of their knot type.