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Wave breaking for a shallow water equation. (English) Zbl 1106.35070

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.

MSC:

35Q35 PDEs in connection with fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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