American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system
  • PROC Home
  • This Issue
  • Next Article
    Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion
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.

[1]

Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507
[2]

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[3]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[4]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[5]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[6]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[7]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311
[8]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[9]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056
[10]

Boling Guo, Zhengde Dai. Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 783-793. doi: 10.3934/dcds.1998.4.783
[11]

Hans Zwart, Yann Le Gorrec, Bernhard Maschke. Relating systems properties of the wave and the Schrödinger equation. Evolution Equations and Control Theory, 2015, 4 (2) : 233-240. doi: 10.3934/eect.2015.4.233
[12]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075
[13]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[14]

Michael Stich, Carsten Beta. Standing waves in a complex Ginzburg-Landau equation with time-delay feedback. Conference Publications, 2011, 2011 (Special) : 1329-1334. doi: 10.3934/proc.2011.2011.1329
[15]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[16]

Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455
[17]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic and Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57
[18]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575
[19]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233
[20]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109

 Impact Factor: 

Tools

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy