The Hopf or inviscid Burgers equation,

u_t+uu_x=0

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.

### References

See 1304.6513 for references and further details.