Nonlinear Dynamics of Two-Dimensional Undercompressive Shocks

Designer shocks for carving out microscale surface morphologies.

Shockwaves are propagating disturbances with a long history of study in gas dynamics, fluid dynamics, and astrophysics. We also see examples of shocks in everyday life such as a traffic jam, in which the oncoming traffic has much lower density than the cars within the traffic jam. A classical compression wave involves propagation of a discontinuity in which information is absorbed from both sid...

Designing steep, sharp patterns on uniformly ion-bombarded surfaces.

We propose and experimentally test a method to fabricate patterns of steep, sharp features on surfaces, by exploiting the nonlinear dynamics of uniformly ion-bombarded surfaces. We show via theory, simulation, and experiment that the steepest parts of the surface evolve as one-dimensional curves that move in the normal direction at constant velocity. The curves are a special solution to the non...

Undercompressive Shocks for a System of Hyperbolic Conservation Laws with Cubic Nonlinearity

KINETIC FUNCTIONS IN MAGNETOHYDRODYNAMICS WITH RESISTIVITY AND HALL EFFECT Dedicated to Professor James Glimm on the occasion of his 75th birthday

We consider a nonlinear hyperbolic system of two conservation laws which arises in ideal magnetohydrodynamics and includes second-order terms accounting for magnetic resistivity and Hall effect. We show that the initial value problem for this model may lead to solutions exhibiting complex wave structures, including undercompressive nonclassical shock waves. We investigate numerically the subtle...

Nonclassical multidimensional viscous and inviscid shocks

Extending our earlier work on Lax-type shocks of systems of conservation laws, we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Laxor undercompressive-type shock waves of nonconservative hyperbolic systems with parabolic regularization. The hyperbolic equations may be of variable multiplicity and the parabolic regularizati...