### 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) |
---|---|

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

Volume | 98 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1985 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

**An invariant measure for the equation u _{tt}-u_{xx}+u^{3}=0.** / Friedlander, Leonid.

Research output: Contribution to journal › Article

_{tt}-u

_{xx}+u

^{3}=0',

*Communications in Mathematical Physics*, vol. 98, no. 1, pp. 1-16. https://doi.org/10.1007/BF01211041

}

TY - JOUR

T1 - An invariant measure for the equation utt-uxx+u3=0

AU - Friedlander, Leonid

PY - 1985/3

Y1 - 1985/3

N2 - Numerical studies of the initial boundary-value problem of the semilinear wave equation utt-uxx+u3=0 subject to periodic boundary conditions u(t, 0)=u(t, 2 π), ut(t, 0)=ut(t, 2 π) and initial conditions u(0, x)=u0(x), ut(0, x)=v0(x), where u0(x) and v0(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial state u0(x), v0(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.

AB - Numerical studies of the initial boundary-value problem of the semilinear wave equation utt-uxx+u3=0 subject to periodic boundary conditions u(t, 0)=u(t, 2 π), ut(t, 0)=ut(t, 2 π) and initial conditions u(0, x)=u0(x), ut(0, x)=v0(x), where u0(x) and v0(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial state u0(x), v0(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.

UR - http://www.scopus.com/inward/record.url?scp=5844225133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=5844225133&partnerID=8YFLogxK

U2 - 10.1007/BF01211041

DO - 10.1007/BF01211041

M3 - Article

AN - SCOPUS:5844225133

VL - 98

SP - 1

EP - 16

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -