Uniqueness of weak solutions with compacton initial data

Establish uniqueness of weak solutions of the gradient-power dispersive equation with compacton initial data, including in the edge-cusped regime where the travelling-wave profile is not classical.

Background

The paper proves that the explicitly constructed compacton, extended by zero, is a distributional weak solution even when its derivative diverges at the free boundary and its second derivative is singular at the crest. This result concerns only the travelling wave itself and does not establish uniqueness for the corresponding initial-value problem.

The authors note that uniqueness is already delicate for the Rosenau–Hyman equation K(p,p). Determining whether compacton initial data generate a unique weak evolution for the two-exponent gradient-power family is therefore an unresolved question.