Wave breaking for a shallow water equation
The author studies the shallow water equation \[ u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx},\quad t> 0,\quad x\in\mathbb R, \] \[ u(x,0)= u_0(x),\qquad x\in \mathbb R, \] 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 \(u_0(x)\) to guarantee the wave breaking. The conditions are written out as explicit integral or differential inequalities.
scientific article; zbMATH DE number 2105764