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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. [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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