### Abstract

Numerical studies of the initial boundary-value problem of the semilinear wave equation u_{tt}-u_{xx}+u^{3}=0 subject to periodic boundary conditions u(t, 0)=u(t, 2 π), u_{t}(t, 0)=u_{t}(t, 2 π) and initial conditions u(0, x)=u_{0}(x), u_{t}(0, x)=v_{0}(x), where u_{0}(x) and v_{0}(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial state u_{0}(x), v_{0}(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above "returning" property.

Original language | English (US) |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

Volume | 98 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1985 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics