The dominant instability of near-extreme Stokes waves.

The instability of Stokes waves, steady propagating waves on the surface of an ideal fluid of infinite depth, is a fundamental problem in the field of nonlinear science. The dominant instability of these waves depends on their steepness. For small amplitude waves, it is well known that the Benjamin-Feir or modulational instability dominates the dynamics of a wave train. We demonstrate that for steeper waves, an instability caused by disturbances localized at the wave crest vastly surpasses the growth rate of the modulational instability. These dominant localized disturbances are either coperiodic with the Stokes wave or have twice its period. In either case, the nonlinear evolution of the instability leads to the formation of plunging breakers. This phenomenon explains why long propagating ocean swell consists of small-amplitude waves.

Neck instability of bright solitons in normally dispersive Kerr media.

The neck instability of bright solitons of the hyperbolic nonlinear Shrödinger equation is investigated. It is shown that this instability originates from a four-wave mixing interaction that links on-axis to off-axis radiation at opposite frequency bands. Our experiment supports this interpretation.

Non-steady-state heat conduction in composite walls.

The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but neither temperature nor heat flux is prescribed there. Instead, the physical assumptions of their continuity at the interfaces are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. We indicate also how to extend the solution method to the setting of one finite-sized domain surrounded on both sides by semi-infinite domains, and on that of three finite-sized domains.