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**On the quintic nonlinear Schrödinger equation created by the vibrations of a square plate on a weakly nonlinear elastic foundation and the stability of the uniform solution.**
*(English)*
Zbl 1159.74016

The authors consider an envelope surface created by the vibrations of a square plate on a weakly nonlinear elastic foundation. The stability of the uniform solution of the governing equation for the envelope surface is analyzed. Additionally, the authors derive a two-dimensional equation that governs the spatial and temporal evolution of the envelope surface on the cubic nonlinear elastic foundation. It is shown that the governing equation becomes the quintic nonlinear Schrödinger equation. The stability condition for the uniform solution of the quintic nonlinear Schrödinger equation has been obtained.

Reviewer: Girish Ramaiah (Bangalore)

### MSC:

74H45 | Vibrations in dynamical problems in solid mechanics |

74K20 | Plates |

35Q55 | NLS equations (nonlinear Schrödinger equations) |

### Software:

COMSOL
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\textit{B. T. Nohara} and \textit{A. Arimoto}, Japan J. Ind. Appl. Math. 24, No. 2, 161--179 (2007; Zbl 1159.74016)

Full Text:
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### References:

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