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Abstract
Scalar conservation laws, such as Burgers equation, can develop discontinuities that propagate as nonlinear waves such as shocks or rarefaction waves. While the space of such weak solutions is huge, adding higher-order physics to the problem can vastly restrict the possible solutions. For example, adding viscosity forces the shocks to be "compressive" (they satisfy the Lax entropy condition), which means there is a restriction on the shock speed. However, for many physical systems the appropriate regularization is not viscosity, but rather a higher-order term such as surface diffusion. In these systems there may exist shocks that violate the Lax entropy condition -- so-called "undercompressive" shocks -- and these have dynamical properties that can be quite different from compressive shocks.
We will look at a particular system where undercompressive shocks are important, namely pattern formation on surfaces that are eroded by ion bombardment. We will look at the set of undercompressive shocks in the one- and two-dimensional equations, and show that these have striking dynamical consequences that may eventually be useful for making nano-scale patterns on surfaces. |
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