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2013, 2013(special): 415-426. doi: 10.3934/proc.2013.2013.415

The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion

1. 

Department of Mathematics, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan

2. 

Department of Information Science, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan

Received  September 2012 Revised  April 2013 Published  November 2013

We first survey the two-dimensional governing equation that describes the propagation of a wave packet on an elastic plate using the method of multiple scales by [13]. We next expand the governing equation to the multi-dimensional case not only in the sense of mathematical science but also engineering.
Citation: Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415
References:
[1]

G. P. Agrawal, "Fiber-Optic Communication System," 2nd editon, Wiley, New York, 1997. Google Scholar

[2]

R. C. Averill and J. N. Reddy, Behavior of plate elements based on the first-order shear deformation theory, Engineering Computations, 7 (1990), 57-74. Google Scholar

[3]

S. K. Chakrabarti, R. H. Snider and P. H. Feldhausen, Mean length of runs of ocean waves, Journal of Geophysical Research, 79(1974), 5665-5667. Google Scholar

[4]

H. N. Chu and G. Herrmann, Influence of large amplitudes on free flexural vibrations of rectangular elastic plates, Journal of Applied Mechnics, 23 (1956), 532-540.  Google Scholar

[5]

Y. Goda, Numerical experiments on wave statistics with spectral simulation, Report Port Harbour Research Institute, 9 (1970), 3-57. Google Scholar

[6]

R. Haberman, "Elementary Applied Partial Differential Equations," Prentice Hall, Englewood Cliff, NJ, 1983.  Google Scholar

[7]

M .M. Hrabok and T. M. Hrudey, A review and catalog of plate bending finite elements, Computers and Structures, 19 (1984), 479-495. Google Scholar

[8]

S. Kanagawa, K. Tchizawa and T. Nitta, Solutions of Ginzburg-Landau Equations Induced from Multi-dimensional Bichromatic Waves and Some Examples of Their Envelope Functions, Theoretical and Applied Mechanics Japan, 58 (2009), 71-78.  Google Scholar

[9]

S. Kanagawa, K. Tchizawa and T. Nitta, Ginzburg-Landau equations induced from multi-dimensional bichromatic waves, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e2258-e2266.  Google Scholar

[10]

S. Kanagawa, T. Nitta and K. Tchizawa, Approximated Solutions of Schrodinger Equations Induced from Nearly Monochromatic Waves, Theoretical and Applied Mechanics Japan, 59 (2010), 153-161. Google Scholar

[11]

A. W. Leissa, "Vibration of Plates," NASA-Sp-160, 1969. Google Scholar

[12]

M. S. Longuet-Higgins, Statistical properties of wave groups in a random sea-state, Philosophical Transactions of the Royal Society of London, Series A, 312(1984), 219-250.  Google Scholar

[13]

A. H. Nayfeh, "Perturbation Methods," Wiley, New York, 2002.  Google Scholar

[14]

B. T. Nohara, Governing Equations of Envelope Surface Created by Directional, Nearly Monochromatic Waves, Journal of Society of Industrial and Applied Mathematics, 13 (2003), 75-86. (in Japanese)  Google Scholar

[15]

B. T. Nohara, Derivation and consideration of governing equations of the envelope surface created by directional, nearly monochromatic waves, International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 31 (2003), 375-392.  Google Scholar

[16]

B. T. Nohara, Governing Equations of Envelope Surface Created by Nearly Bichromatic Waves Propagating on an Elastic Plate and Their Stability, Japan Journal of Industrial and Applied Mathematics, 22 (2005), 87-109.  Google Scholar

[17]

B. T. Nohara and A. Arimoto, The stability of the governing equation of envelope surface created by nearly bichromatic waves propagating on an elastic plate, Nonlinear Analysis: Theory, Methods & Applications, 63 (2005), e2197-e2208. Google Scholar

[18]

B. T. Nohara and A. Arimoto, On the Quintic Nonlinear Schrodinger Equation Created by the Vibrations of a Square Plate on a Weakly Nonlinear Elastic Foundation and the Stability of the Uniform Solution, Japan Journal of Industrial and Applied Mathematics, 24 (2007), 161-179.  Google Scholar

[19]

B. T. Nohara and A. Arimoto and T. Saigo, Governing Equations of Envelopes Created by Nearly Bichromatic Waves and Relation to the Nonlinear Schrödinger Equation, Chaos, Solitons and Fractals, 35 (2008), 942-948.  Google Scholar

[20]

J. N. Reddy, "An Introduction to the Finite Element Method," 2nd edition., McGraw-Hill, New York, 1993. Google Scholar

[21]

H. Reismann, "Elastic Plates: Theory and Application," Wiley, New Jersey, 1988. Google Scholar

[22]

S. P. Timoshenko, "Theory of Plates and Shells," McGraw-Hill, New York, 1940. Google Scholar

[23]

S. P. Timoshenko and S. Woinowsky-Krieger, "Theory of Plates and Shells," McGraw-Hill, Singapore, 1970. Google Scholar

[24]

A. C. Ugural, "Stresses in plates and shells," McGraw-Hill, New York, 1981. Google Scholar

[25]

H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Physics Review Letters, 17 (1966), 996-998. Google Scholar

[26]

M.A. Zarubinskaya and W.T. van Horssen, On the Vibration on a Simply Supported Square Plate on a Weakly Nonlinear Elastic Fooundation, International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 40 (2005), 35-60.  Google Scholar

show all references

References:
[1]

G. P. Agrawal, "Fiber-Optic Communication System," 2nd editon, Wiley, New York, 1997. Google Scholar

[2]

R. C. Averill and J. N. Reddy, Behavior of plate elements based on the first-order shear deformation theory, Engineering Computations, 7 (1990), 57-74. Google Scholar

[3]

S. K. Chakrabarti, R. H. Snider and P. H. Feldhausen, Mean length of runs of ocean waves, Journal of Geophysical Research, 79(1974), 5665-5667. Google Scholar

[4]

H. N. Chu and G. Herrmann, Influence of large amplitudes on free flexural vibrations of rectangular elastic plates, Journal of Applied Mechnics, 23 (1956), 532-540.  Google Scholar

[5]

Y. Goda, Numerical experiments on wave statistics with spectral simulation, Report Port Harbour Research Institute, 9 (1970), 3-57. Google Scholar

[6]

R. Haberman, "Elementary Applied Partial Differential Equations," Prentice Hall, Englewood Cliff, NJ, 1983.  Google Scholar

[7]

M .M. Hrabok and T. M. Hrudey, A review and catalog of plate bending finite elements, Computers and Structures, 19 (1984), 479-495. Google Scholar

[8]

S. Kanagawa, K. Tchizawa and T. Nitta, Solutions of Ginzburg-Landau Equations Induced from Multi-dimensional Bichromatic Waves and Some Examples of Their Envelope Functions, Theoretical and Applied Mechanics Japan, 58 (2009), 71-78.  Google Scholar

[9]

S. Kanagawa, K. Tchizawa and T. Nitta, Ginzburg-Landau equations induced from multi-dimensional bichromatic waves, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), e2258-e2266.  Google Scholar

[10]

S. Kanagawa, T. Nitta and K. Tchizawa, Approximated Solutions of Schrodinger Equations Induced from Nearly Monochromatic Waves, Theoretical and Applied Mechanics Japan, 59 (2010), 153-161. Google Scholar

[11]

A. W. Leissa, "Vibration of Plates," NASA-Sp-160, 1969. Google Scholar

[12]

M. S. Longuet-Higgins, Statistical properties of wave groups in a random sea-state, Philosophical Transactions of the Royal Society of London, Series A, 312(1984), 219-250.  Google Scholar

[13]

A. H. Nayfeh, "Perturbation Methods," Wiley, New York, 2002.  Google Scholar

[14]

B. T. Nohara, Governing Equations of Envelope Surface Created by Directional, Nearly Monochromatic Waves, Journal of Society of Industrial and Applied Mathematics, 13 (2003), 75-86. (in Japanese)  Google Scholar

[15]

B. T. Nohara, Derivation and consideration of governing equations of the envelope surface created by directional, nearly monochromatic waves, International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 31 (2003), 375-392.  Google Scholar

[16]

B. T. Nohara, Governing Equations of Envelope Surface Created by Nearly Bichromatic Waves Propagating on an Elastic Plate and Their Stability, Japan Journal of Industrial and Applied Mathematics, 22 (2005), 87-109.  Google Scholar

[17]

B. T. Nohara and A. Arimoto, The stability of the governing equation of envelope surface created by nearly bichromatic waves propagating on an elastic plate, Nonlinear Analysis: Theory, Methods & Applications, 63 (2005), e2197-e2208. Google Scholar

[18]

B. T. Nohara and A. Arimoto, On the Quintic Nonlinear Schrodinger Equation Created by the Vibrations of a Square Plate on a Weakly Nonlinear Elastic Foundation and the Stability of the Uniform Solution, Japan Journal of Industrial and Applied Mathematics, 24 (2007), 161-179.  Google Scholar

[19]

B. T. Nohara and A. Arimoto and T. Saigo, Governing Equations of Envelopes Created by Nearly Bichromatic Waves and Relation to the Nonlinear Schrödinger Equation, Chaos, Solitons and Fractals, 35 (2008), 942-948.  Google Scholar

[20]

J. N. Reddy, "An Introduction to the Finite Element Method," 2nd edition., McGraw-Hill, New York, 1993. Google Scholar

[21]

H. Reismann, "Elastic Plates: Theory and Application," Wiley, New Jersey, 1988. Google Scholar

[22]

S. P. Timoshenko, "Theory of Plates and Shells," McGraw-Hill, New York, 1940. Google Scholar

[23]

S. P. Timoshenko and S. Woinowsky-Krieger, "Theory of Plates and Shells," McGraw-Hill, Singapore, 1970. Google Scholar

[24]

A. C. Ugural, "Stresses in plates and shells," McGraw-Hill, New York, 1981. Google Scholar

[25]

H. Washimi and T. Taniuti, Propagation of ion-acoustic solitary waves of small amplitude, Physics Review Letters, 17 (1966), 996-998. Google Scholar

[26]

M.A. Zarubinskaya and W.T. van Horssen, On the Vibration on a Simply Supported Square Plate on a Weakly Nonlinear Elastic Fooundation, International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 40 (2005), 35-60.  Google Scholar

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