Wave breaking for a shallow water equation.

*(English)*Zbl 1106.35070The author studies the shallow water equation

$${u}_{t}-{u}_{xxt}+3u{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx},\phantom{\rule{1.em}{0ex}}t>0,\phantom{\rule{1.em}{0ex}}x\in \mathbb{R},$$

$$u(x,0)={u}_{0}\left(x\right),\phantom{\rule{2.em}{0ex}}x\in \mathbb{R},$$

which can be considered as a bi-Hamiltonian generalization of Korteweg-de Vries equations. The wave is said to be broken, if the solution $u$ remains bounded, but its slope becomes infinite in finite time. The author formulates three various sufficient conditions on initial data ${u}_{0}\left(x\right)$ to guarantee the wave breaking. The conditions are written out as explicit integral or differential inequalities.

Reviewer: Oleg Titow (Berlin)