### Abstract

It is shown that the equations of motion of an ideal fluid with a free surface in the absence of both gravitational and capillary forces can be effectively solved in the approximation of small surface angles. It can be done by means of an analytical continuation of both the velocity potential on the surface and its elevation. For almost arbitrary initial conditions the system evolves to the formation of singularities in a finite time. Three kinds of singularities are shown to be possible. The first one is of the root character provided by the analytical behavior of the velocity potential. In this case the process of the singularity formation, representing some analog of the wave breaking, is described as a motion of branch points in the complex plane towards the real axis. The second type can be obtained as a result of the interaction of two movable branch points leading to the formation of wedges on the free surface. The third kind is associated with a motion in the complex plane of the singular points of the analytical continuation of the elevation, resulting in the appearance of strong singularities for the surface profile.

Original language | English (US) |
---|---|

Pages (from-to) | 1283-1290 |

Number of pages | 8 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - 1994 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*49*(2), 1283-1290. https://doi.org/10.1103/PhysRevE.49.1283

**Formation of singularities on the free surface of an ideal fluid.** / Kuznetsov, E. A.; Spector, M. D.; Zakharov, Vladimir E.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 49, no. 2, pp. 1283-1290. https://doi.org/10.1103/PhysRevE.49.1283

}

TY - JOUR

T1 - Formation of singularities on the free surface of an ideal fluid

AU - Kuznetsov, E. A.

AU - Spector, M. D.

AU - Zakharov, Vladimir E

PY - 1994

Y1 - 1994

N2 - It is shown that the equations of motion of an ideal fluid with a free surface in the absence of both gravitational and capillary forces can be effectively solved in the approximation of small surface angles. It can be done by means of an analytical continuation of both the velocity potential on the surface and its elevation. For almost arbitrary initial conditions the system evolves to the formation of singularities in a finite time. Three kinds of singularities are shown to be possible. The first one is of the root character provided by the analytical behavior of the velocity potential. In this case the process of the singularity formation, representing some analog of the wave breaking, is described as a motion of branch points in the complex plane towards the real axis. The second type can be obtained as a result of the interaction of two movable branch points leading to the formation of wedges on the free surface. The third kind is associated with a motion in the complex plane of the singular points of the analytical continuation of the elevation, resulting in the appearance of strong singularities for the surface profile.

AB - It is shown that the equations of motion of an ideal fluid with a free surface in the absence of both gravitational and capillary forces can be effectively solved in the approximation of small surface angles. It can be done by means of an analytical continuation of both the velocity potential on the surface and its elevation. For almost arbitrary initial conditions the system evolves to the formation of singularities in a finite time. Three kinds of singularities are shown to be possible. The first one is of the root character provided by the analytical behavior of the velocity potential. In this case the process of the singularity formation, representing some analog of the wave breaking, is described as a motion of branch points in the complex plane towards the real axis. The second type can be obtained as a result of the interaction of two movable branch points leading to the formation of wedges on the free surface. The third kind is associated with a motion in the complex plane of the singular points of the analytical continuation of the elevation, resulting in the appearance of strong singularities for the surface profile.

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U2 - 10.1103/PhysRevE.49.1283

DO - 10.1103/PhysRevE.49.1283

M3 - Article

VL - 49

SP - 1283

EP - 1290

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 2

ER -