Kinks and the minimal surface equation in Minkowski space (Q917127)

The scalar field u is defined on 4-D Minkowski space, and satisfies the nonlinear wave equation \[ \epsilon^ 2\square u-V'(u)=0. \] V(u) has minima at \(u=0\), 1 with \(V=0\). In the limit \(\epsilon\to 0\) there are solutions in which regions with \(u=0\) or \(u=1\) are separated by layers of thickness \(\epsilon\) about 3-D timelike surfaces, called kinks. The author shows by singular perturbation analysis that the kink surfaces satisfy a minimal surface equation in Minkowski space. He shows that a kink with a microstructure of small ripples is stable against collapse. All of the results in this interesting paper are established by a simple generalization of Whitham's averaged Lagrangian method [\textit{G. Whitham}, Linear and Nonlinear Waves (Wiley, New York, 1974; Zbl 0373.76001)].