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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

Shuya Kanagawa 1, and  Ben T. Nohara 2,

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.
Keywords: bichromatic wave., elastic plate, envelope, Schrödinger equation, Ginzburg-Landau equation, governing equation, monochromatic wave.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35Q5.
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.

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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)

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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)

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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.

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