# Shock waves and compactons for fifth-order non-linear dispersion equations

### Reference:

Galaktionov, V., 2010. Shock waves and compactons for fifth-order non-linear dispersion equations. *European Journal of Applied Mathematics*, 21 (1), pp. 1-50.

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### Official URL:

http://dx.doi.org/10.1017/S0956792509990118

### Abstract

The following first problem is posed - to justify that the standing shock wave S-(x) = -sign x ={-1 for x < 0, 1 for x > 0 F6 r x < 0, is a correct 'entropy solution' of the Cauchy problem for the fifth-ordcr degenerate non-linear dispersion equations (NDEs), as for the classic Euler one u(1) + uu(x) = 0, u(1) = -(uu(x))(xxxx) and u(1) = -(uu(xxx))(x) m R x R+. These two quasi-linear degenerate partial differential equations (PDEs) arc chosen as typical representatives, so other (2m + 1)th-order NDEs of non-divergent form admit such shocks waves. As a related second problem, the opposite initial shock S+(x) = -S-(x) = sign x is shown to be a non-entropy solution creating a rarefaction wave, which becomes C-infinity for any t > 0 Formation of shocks leads to non-uniqueness of any 'entropy solutions'. Similar phenomena are studied for a fifth-order in time NDE u(uuu) = (uu(x))(xxxx) in normal form On the other hand, related NDEs, such as u(t) = -(|u|u(x))(xxxx) + |u|u(x) in R x R+, are shown to admit smooth compactons, as oscillatory travelling wave solutions with compact support. The well-known non-negative compactons, which appeared in various applications (first examples by Dcy 1998, Phys Rev E, vol. 57, pp 4733-4738, and Rosenau and Levy, 1999, Phys Lett. A, vol. 252, pp 297-306), are non-existent in general and are not robust relative to small perturbations of parameters of the PDE.

### Details

Item Type | Articles |

Creators | Galaktionov, V. |

DOI | 10.1017/s0956792509990118 |

Departments | Faculty of Science > Mathematical Sciences |

Publisher Statement | galaktionov-v-a.pdf: © Cambridge University Press Published in European Journal of Applied Mathematics. Galaktionov, V., 2010. Shock waves and compactons for fifth-order non-linear dispersion equations. European Journal of Applied Mathematics, 21 (1), pp. 1-50. This version is subject to a 12 month embargo from date of publication. Published version available from: http://dx.doi.org/10.1017/S0956792509990118 |

Refereed | Yes |

Status | Published |

ID Code | 17986 |

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