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, (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.   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.   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.   Google Scholar

[5]

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

[6]

R. Haberman, "Elementary Applied Partial Differential Equations,", Prentice Hall, (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.   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.   Google Scholar

[9]

S. Kanagawa, K. Tchizawa and T. Nitta, Ginzburg-Landau equations induced from multi-dimensional bichromatic waves,, Nonlinear Analysis: Theory, 71 (2009).   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.   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, 312 (1984), 219.   Google Scholar

[13]

A. H. Nayfeh, "Perturbation Methods,", Wiley, (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.   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.   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.   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, 63 (2005).   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.   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, 35 (2008), 942.   Google Scholar

[20]

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

[21]

H. Reismann, "Elastic Plates: Theory and Application,", Wiley, (1988).   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

show all references

References:
[1]

G. P. Agrawal, "Fiber-Optic Communication System,", 2nd editon, (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.   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.   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.   Google Scholar

[5]

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

[6]

R. Haberman, "Elementary Applied Partial Differential Equations,", Prentice Hall, (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.   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.   Google Scholar

[9]

S. Kanagawa, K. Tchizawa and T. Nitta, Ginzburg-Landau equations induced from multi-dimensional bichromatic waves,, Nonlinear Analysis: Theory, 71 (2009).   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.   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, 312 (1984), 219.   Google Scholar

[13]

A. H. Nayfeh, "Perturbation Methods,", Wiley, (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.   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.   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.   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, 63 (2005).   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.   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, 35 (2008), 942.   Google Scholar

[20]

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

[21]

H. Reismann, "Elastic Plates: Theory and Application,", Wiley, (1988).   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

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

[1]

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

Boling Guo, Zhengde Dai. Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 783-793. doi: 10.3934/dcds.1998.4.783
[3]

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

N. Maaroufi. Topological entropy by unit length for the Ginzburg-Landau equation on the line. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 647-662. doi: 10.3934/dcds.2014.34.647
[5]

Jingna Li, Li Xia. The Fractional Ginzburg-Landau equation with distributional initial data. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2173-2187. doi: 10.3934/cpaa.2013.12.2173
[6]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[7]

Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665
[8]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[9]

Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311
[10]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[11]

Hongzi Cong, Jianjun Liu, Xiaoping Yuan. Quasi-periodic solutions for complex Ginzburg-Landau equation of nonlinearity $|u|^{2p}u$. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 579-600. doi: 10.3934/dcdss.2010.3.579
[12]

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
[13]

Boling Guo, Bixiang Wang. Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 455-466. doi: 10.3934/dcds.1996.2.455
[14]

N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934/dcds.2007.19.711
[15]

Yueling Jia, Zhaohui Huo. Inviscid limit behavior of solution for the multi-dimensional derivative complex Ginzburg-Landau equation. Kinetic & Related Models, 2014, 7 (1) : 57-77. doi: 10.3934/krm.2014.7.57
[16]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575
[17]

Hong Lu, Shujuan Lü, Mingji Zhang. Fourier spectral approximations to the dynamics of 3D fractional complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2539-2564. doi: 10.3934/dcds.2017109
[18]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[19]

O. Goubet, N. Maaroufi. Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1253-1267. doi: 10.3934/cpaa.2012.11.1253
[20]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

 Impact Factor: 

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences