Japan Journal of Industrial and Applied Mathematics

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

Ben T. Nohara and Akio Arimoto

Full-text: Open access

Abstract

Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures. The study of plates from theoretical perspective as well as experimental viewpoint is fundamental to understanding of the behavior of such structures. The dynamic characteristics of plates, such as natural vibrations, transient responses for the external forces and so on, are especially of importance in actual environments. In this paper, we conside the envelope surface created by the vibrations of a square plate on a weakly nonliner elastic foundation and analyze the stability of the uniform solution of the governing equation for the envelope surface. We derive the two-dimensional equation that governs the spatial and temporal evolution of the envelope surface on cubic nonlinear elastic foundation. The fact that the governing equation becomes the quintic nonlinear Schrödinger equation is shown. Also we obtain the stability condition of the uniform solution of the quintic nonlinear Schrödinger equation.

Article information

Source
Japan J. Indust. Appl. Math., Volume 24, Number 2 (2007), 161-179.

Dates
First available in Project Euclid: 17 December 2007

Permanent link to this document
https://projecteuclid.org/euclid.jjiam/1197908778

Mathematical Reviews number (MathSciNet)
MR2338152

Zentralblatt MATH identifier
1159.74016

Keywords
elastic foundation envelope nearly monochromatic waves perturbation Schrödinger equation

Citation

Nohara, Ben T.; Arimoto, Akio. 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. Japan J. Indust. Appl. Math. 24 (2007), no. 2, 161--179. https://projecteuclid.org/euclid.jjiam/1197908778


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