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This issue’s SIGEST paper from the SIAM Journal on Applied Dynamical Systems, “Us-ing Global Invariant Manifolds to Understand Metastability in the Burgers Equation withSmall Viscosity,” by Margaret Beck and C. Eugene Wayne, is about the behavior of thesolutions of dynamical systems (i.e., systems of time-dependent differential equations,either ordinary or partial) with small dissipation.

Dynamical systems describe the world around us; a classic example is the motionof a fluid. When these systems are forced, coherent structures often emerge such as atsunami in the aftermath of an earthquake or the vortex produced by a plane’s wingtipor a canoe paddle. These states are metastable—they persist for a long time, althougheventually the viscous forces of the fluid lead them to dissipate.

The SIGEST paper is about metastable solutions to the Burgers equation. TheBurgers equation is a partial differential equation that arises in models of gas dy-namics, traffic flow, and qualitatively captures some important features of the two-dimensional Navier–Stokes equations for fluid motion. The authors show rigorouslythat for the Burgers equation with small dissipation, most initial conditions first ap-proach a metastable N-wave solution (which, not surprisingly, is shaped like the letterN). Eventually this state dissipates and the solution approaches a self-similar diffusivewave, which then decays to a constant solution.

The authors present a lucid analysis by developing rigorous asymptotics for thesmall viscosity limit. They characterize the solutions’ approach to global invariant man-ifolds, corresponding to the metastable states, and do a fine job of explaining thischallenging mathematics.

The Editors

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