  In an influential 1964 article, P. Lax studied $2 \times 2$ genuinely
nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the
method of Riemann invariants, he showed that a large set of smooth initial data
lead to bounded solutions whose first spatial derivatives blow up in finite
time, a phenomenon known as wave breaking. In the present article, we study the
Cauchy problem for two classes of quasilinear wave equations in two spatial
dimensions that are closely related to the systems studied by Lax. When the
data have one-dimensional symmetry, Lax's methods can be applied to the wave
equations to show that a large set of smooth initial data lead to wave
breaking. Here we study solutions with initial data that are close, as measured
by an appropriate Sobolev norm, to data belonging to a distinguished subset of
Lax's data: the data corresponding to simple plane waves. Our main result is
that under suitable relative smallness assumptions, the Lax-type wave breaking
for simple plane waves is stable. The key point is that we allow the data
perturbations to break the symmetry. Moreover, we give a detailed, constructive
description of the asymptotic behavior of the solution all the way up to the
first singularity, which is a shock driven by the intersection of null
(characteristic) hyperplanes. We also outline how to extend our results to the
compressible irrotational Euler equations. To derive our results, we use
Christodoulou's framework for studying shock formation to treat a new solution
regime in which wave dispersion is not present.
