  We prove that any simple planar travelling wave solution to the membrane
equation in spatial dimension $d \geq 3$ with bounded spatial extent is
globally nonlinearly stable under sufficiently small compactly-supported
perturbations, where the smallness depends on the size of the support of the
perturbation as well as on the initial travelling wave profile. The main
novelty of the argument is the lack of higher-order peeling in our vector-field
based method. In particular, the higher order energies (in fact, all energies
at order $2$ or higher) are allowed to grow polynomially (but in a controlled
way) in time. This is in contrast with classical global stability arguments
where only the "top" order energies used in the bootstrap argument exhibit
growth, and reflects the fact that the background travelling wave solution has
"infinite energy" and the coefficients of the perturbation equation are not
asymptotically Lorentz invariant. Nonetheless, we can prove that the
perturbation converges to zero in $C^2$ by carefully analyzing the nonlinear
interactions and exposing a certain "vestigial" null structure in the
equations.
