The Hopf or inviscid Burgers equation,
is one of the simplest models for wave phenomena showing wave breaking, i.e., the appearance of a gradient catastrophe (an orthogonal tangent) for generic localised initial data. A gradient catastrophe indicates the onset of a discontinuity of the solution, the formation of a shock.
The video shows the solution via the method of characteristics for the initial data
u(x,0) = sech^2(x)
which, after the gradient catastrophe, gives a multivalued solution, one of an infinite number of weak solutions to the Hopf equation for these initial data.
See 1304.6513 for references and further details.