*(English)*Zbl 1102.35021

The nonlinear dispersive equation

is, in dimensionless space-time variable $(x,t),$ a model for unidirectional shallow water waves over a flat bottom. It was derived using asymptotic expansions directly in the Hamiltonian for Euler’s equations in the shallow water regime, where $u(t,x)$ represents the fluid velocity.

For $\gamma /{c}_{0}<0$, short waves and long waves travel in the same direction. Using the notation $m=u-{\alpha}^{2}{u}_{xx}$, it is possible to rewrite the equation (1) as

the constants ${\alpha}^{2}$ and $\gamma /{c}_{0}$ are squares of length scales, and the constant ${c}_{0}=2\omega =\sqrt{gh}>0$ is the critical shallow water speed for undisturbed water at rest at spatial infinity, where $h$ is the mean fluid depth and $g$ the gravitational constant, $9\xb78m/{s}^{2}$, $\omega >0,\phantom{\rule{4pt}{0ex}}\alpha >0\xb7$

Equation (1.2) is connected with two separately integrable soliton equations for shallow water waves. Formally, when ${\alpha}^{2}=0$, (2) becomes the Korteweg-de Vries (KdV) equation

Hence, when $\omega =0$, it is well known that there exists a smooth soliton

Instead, when $\gamma =0$ in (2), it turns out to be the Camassa-Holm (CH) equation

It is shown in this paper, that in the case $\gamma =-2\omega {\alpha}^{2},$ either there exist global solutions for (2), or breaking wave occurs in finite time for certain initial profiles. The author found the exact rate of breaking waves for a class of initial profiles. Moreover, lower bounds of the existence time of solutions to (2) are determined too.

The formal approach to prove global existence or wave breaking for shallow water wave (1.2) in this paper originates in an idea of *A. Constantin* [Ann. Inst. Fourier 50, 321–362 (2000; Zbl 0944.35062)]. That is, it is shown that for a large class of initial profiles the corresponding solutions to (1.2) either exist globally in time or blow up in finite time by using a continuous family of diffeomorphisms of the line associated to (2).

The plan of the paper is as follows. In Section 2, the existence of global solutions is studied. Section 3 turns to investigate the phenomenon of blow-up and describe in detail the wave-breaking mechanism of solutions for (2). Section 4 is devoted to find the rate of blow-up for certain initial profiles. In the last section, Section 5, lower bound for the maximal time of existence of solutions to (2) is determined.

##### MSC:

35B40 | Asymptotic behavior of solutions of PDE |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76B03 | Existence, uniqueness, and regularity theory (fluid mechanics) |

35Q53 | KdV-like (Korteweg-de Vries) equations |

35Q35 | PDEs in connection with fluid mechanics |

35Q51 | Soliton-like equations |

76B25 | Solitary waves (inviscid fluids) |

##### Keywords:

nonlinear dispersive equation; shallow water waves; integrable soliton equations; Korteweg-de Vries equation; Camassa-Holm equation; existence of global solutions; breaking wave; rate of blow-up##### References:

[1] | Beals, R., Sattinger, D., Szmigielski, J.: Inverse scattering and some finite-dimensional integrable systems. Recent developments in integrable systems and Riemann-Hilbert problems (Birmingham, AL, 2000). Contemp. Math. 326, 23–31 (2003) |

[2] | Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993) · Zbl 0936.35153 · doi:10.1103/PhysRevLett.71.1661 |

[3] | Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier, 50, 321–362 (2000) |

[4] | Constantin, A.: On the scattering problem for the Camassa-Holm equation. Proc. R. Soc. Lond. A457, 953–970 (2001) |

[5] | Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998) · Zbl 0923.76025 · doi:10.1007/BF02392586 |

[6] | Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53, 603–610 (2000) · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L |

[7] | Constantin, A., Strauss, W.: Stability of the Camassa-Holm solitons. J. Nonlinear Sci. 12, 415–422 (2002) · Zbl 1022.35053 · doi:10.1007/s00332-002-0517-x |

[8] | Constantin, A., Molinet, L.: Orbital stability of the solitary waves for a shallow water equation. Phys. D. 157, 75–89 (2001) · doi:10.1016/S0167-2789(01)00298-6 |

[9] | Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. (1) 211, 45–61 (2000) |

[10] | Danchin, R.: A few remarks on the Camassa-Holm equation, Differential Integral Equations. 14, 953–988 (2001) |

[11] | Dullin, H., Gottwald, G., Holm, D.: An integrable shallow water equation with linear and nonlinear dispersion. Phy. Rev. Lett. 87(19), 194501(4) (2001) · doi:10.1103/PhysRevLett.87.194501 |

[12] | Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations. Springer Lecture Notes in Math. 448, 25–70 (1975) · doi:10.1007/BFb0067080 |

[13] | McKean, H.: Integrable systems and algebraic curves, Global Analysis(Calgary, 1978), Springer Lecture Notes in Math. 755, 83–200 (1979) |

[14] | Rodriguez-Blanco, G.: On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal. 46, 309–327 (2001) |

[15] | Strauss, W.: Nonlinear wave equations. CBMS Regional Conf. Ser. Math., Amer. Math. Soc., Providence, RI 73 (1989) |

[16] | Tian, L.X., Gui, G., Liu, Y.: On the Cauchy problem and the scattering problem for the Dullin-Gottwald-Holm equation. to appear |