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doi: 10.3934/dcdsb.2021075

Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori

Bochao Chen 1,2, and  Yixian Gao 1,,

1. 

School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024
2. 

China, School of College of Mathematics, Jilin University, Changchun, Jilin 130012, China

* Corresponding author: Yixian Gao

Received  August 2020 Revised  January 2021 Early access  March 2021

Fund Project: The first author is supported by NSFC grant 11901232 and China Postdoctoral Science Foundation Funded Projects 2019M651191, 2020T130243. The second author is supported by NSFC grants 11871140, 12071065, JJKH 20180006KJ and FRFCU 2412019BJ005

This paper concerns the mathematical analysis of quasi-periodic travelling wave solutions for beam equations with damping on 3-dimensional rectangular tori. Provided that the generators of the rectangular torus satisfy certain relationships, by excluding some values of two model parameters, we establish the existence of small amplitude quasi-periodic travelling wave solutions with three frequencies. Moreover, it can be shown that such solutions are either continuations of rotating wave solutions, or continuations of quasi-periodic travelling wave solutions with two frequencies, and that the set of two model parameters is dense in the positive quadrant.

Keywords: Beam equations, quasi-periodic solutions, travelling waves, rectangular tori.
Mathematics Subject Classification: 35L75, 37L10.
Citation: Bochao Chen, Yixian Gao. Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021075
References:
[1]

M. Berti, L. Franzoi and A. Maspero, Traveling quasi-periodic water waves with constant vorticity, preprint, arXiv: 2004.08905. Google Scholar

[2]

J. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, in Mathematical Aspects of Nonlinear Dispersive Equations, volume 163 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, (2007), 1–20.  Google Scholar

[3]

S. A. Campbell, J. Bélair, T. Ohira and J. Milton, Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, Chaos, 5 (1995), 640-645.  doi: 10.1063/1.166134.  Google Scholar

[4]

S. A. Campbell, J. Bélair, T. Ohira and J. Milton, Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback, J. Dynam. Differential Equations, 7 (1995), 213-236.  doi: 10.1007/BF02218819.  Google Scholar

[5]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1981/82), 433-454.  doi: 10.1090/qam/644099.  Google Scholar

[6]

Y. Deng, On growth of Sobolev norms for energy critical NLS on irrational tori: Small energy case, Comm. Pure Appl. Math., 72 (2019), 801-834.  doi: 10.1002/cpa.21797.  Google Scholar

[7]

Y. Deng and P. Germain, Growth of solutions to NLS on irrational tori, Int. Math. Res. Not. IMRN, 2019 (2019), 2919-2950.  doi: 10.1093/imrn/rnx210.  Google Scholar

[8]

Y. Deng, P. Germain and L. Guth, Strichartz estimates for the Schrödinger equation on irrational tori, J. Funct. Anal., 273 (2017), 2846-2869.  doi: 10.1016/j.jfa.2017.05.011.  Google Scholar

[9]

R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet–Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.  doi: 10.1016/j.jde.2015.02.043.  Google Scholar

[10]

E. Emmrich and M. Thalhammer, A class of integro-differential equations incorporating nonlinear and nonlocal damping with applications in nonlinear elastodynamics: Existence via time discretization, Nonlinearity, 24 (2011), 2523-2546.  doi: 10.1088/0951-7715/24/9/008.  Google Scholar

[11]

R. Feola and F. Giuliani, Quasi-periodic traveling waves on an infinitely deep fluid under gravity, preprint, arXiv: 2005.08280. Google Scholar

[12]

L. Herrmann, Vibration of the Euler–Bernoulli beam with allowance for dampings, in Proc. World Congr. Eng., London, UK, (2008), 901–904. Google Scholar

[13]

R. Imekraz, Long time existence for the semi-linear beam equation on irrational tori of dimension two, Nonlinearity, 29 (2016), 3067-3102.  doi: 10.1088/0951-7715/29/10/3067.  Google Scholar

[14]

A. Jenkins, Self-oscillation, Phys. Rep., 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007.  Google Scholar

[15]

F. Kogelbauer and G. Haller, Rigorous model reduction for a damped-forced nonlinear beam model: An infinite-dimensional analysis, J. Nonlinear Sci., 28 (2018), 1109-1150.  doi: 10.1007/s00332-018-9443-4.  Google Scholar

[16]

N. Kosovalić, Quasi-periodic self-excited travelling waves for damped beam equations, J. Differential Equations, 265 (2018), 2171-2190.  doi: 10.1016/j.jde.2018.04.022.  Google Scholar

[17]

N. Kosovalić and B. Pigott, Self-excited vibrations for damped and delayed 1-dimensional wave equations, J. Dynam. Differential Equations, 31 (2019), 129-152.  doi: 10.1007/s10884-018-9654-2.  Google Scholar

[18]

K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.  Google Scholar

[19]

A. Longtin and J. G. Milton, Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations, Bull. Math. Biol., 51 (1989), 605-624.   Google Scholar

[20]

D. Takács and G. Stépán, Experiments on quasiperiodic wheel shimmy, J. Comput. Nonlinear Dynam., 4 (2009), 031007. Google Scholar

[21]

H. K. Wang and G. Chen, Asymptotic locations of eigenfrequencies of Euler–Bernoulli beam with nonhomogeneous structural and viscous damping coefficients, SIAM J. Control Optim., 29 (1991), 347-367.  doi: 10.1137/0329019.  Google Scholar

[22]

W. Weaver, Jr., S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, 5$^{nd}$ edition, John Wiley & Sons Limited, 1990. Google Scholar

[23]

J. Wilkening and X. Zhao, Quasi-periodic traveling gravity-capillary waves, preprint, arXiv: 2002.09487. Google Scholar

show all references

References:
[1]

M. Berti, L. Franzoi and A. Maspero, Traveling quasi-periodic water waves with constant vorticity, preprint, arXiv: 2004.08905. Google Scholar

[2]

J. Bourgain, On Strichartz's inequalities and the nonlinear Schrödinger equation on irrational tori, in Mathematical Aspects of Nonlinear Dispersive Equations, volume 163 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, (2007), 1–20.  Google Scholar

[3]

S. A. Campbell, J. Bélair, T. Ohira and J. Milton, Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback, Chaos, 5 (1995), 640-645.  doi: 10.1063/1.166134.  Google Scholar

[4]

S. A. Campbell, J. Bélair, T. Ohira and J. Milton, Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback, J. Dynam. Differential Equations, 7 (1995), 213-236.  doi: 10.1007/BF02218819.  Google Scholar

[5]

G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math., 39 (1981/82), 433-454.  doi: 10.1090/qam/644099.  Google Scholar

[6]

Y. Deng, On growth of Sobolev norms for energy critical NLS on irrational tori: Small energy case, Comm. Pure Appl. Math., 72 (2019), 801-834.  doi: 10.1002/cpa.21797.  Google Scholar

[7]

Y. Deng and P. Germain, Growth of solutions to NLS on irrational tori, Int. Math. Res. Not. IMRN, 2019 (2019), 2919-2950.  doi: 10.1093/imrn/rnx210.  Google Scholar

[8]

Y. Deng, P. Germain and L. Guth, Strichartz estimates for the Schrödinger equation on irrational tori, J. Funct. Anal., 273 (2017), 2846-2869.  doi: 10.1016/j.jfa.2017.05.011.  Google Scholar

[9]

R. Denk and R. Schnaubelt, A structurally damped plate equation with Dirichlet–Neumann boundary conditions, J. Differential Equations, 259 (2015), 1323-1353.  doi: 10.1016/j.jde.2015.02.043.  Google Scholar

[10]

E. Emmrich and M. Thalhammer, A class of integro-differential equations incorporating nonlinear and nonlocal damping with applications in nonlinear elastodynamics: Existence via time discretization, Nonlinearity, 24 (2011), 2523-2546.  doi: 10.1088/0951-7715/24/9/008.  Google Scholar

[11]

R. Feola and F. Giuliani, Quasi-periodic traveling waves on an infinitely deep fluid under gravity, preprint, arXiv: 2005.08280. Google Scholar

[12]

L. Herrmann, Vibration of the Euler–Bernoulli beam with allowance for dampings, in Proc. World Congr. Eng., London, UK, (2008), 901–904. Google Scholar

[13]

R. Imekraz, Long time existence for the semi-linear beam equation on irrational tori of dimension two, Nonlinearity, 29 (2016), 3067-3102.  doi: 10.1088/0951-7715/29/10/3067.  Google Scholar

[14]

A. Jenkins, Self-oscillation, Phys. Rep., 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007.  Google Scholar

[15]

F. Kogelbauer and G. Haller, Rigorous model reduction for a damped-forced nonlinear beam model: An infinite-dimensional analysis, J. Nonlinear Sci., 28 (2018), 1109-1150.  doi: 10.1007/s00332-018-9443-4.  Google Scholar

[16]

N. Kosovalić, Quasi-periodic self-excited travelling waves for damped beam equations, J. Differential Equations, 265 (2018), 2171-2190.  doi: 10.1016/j.jde.2018.04.022.  Google Scholar

[17]

N. Kosovalić and B. Pigott, Self-excited vibrations for damped and delayed 1-dimensional wave equations, J. Dynam. Differential Equations, 31 (2019), 129-152.  doi: 10.1007/s10884-018-9654-2.  Google Scholar

[18]

K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.  Google Scholar

[19]

A. Longtin and J. G. Milton, Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations, Bull. Math. Biol., 51 (1989), 605-624.   Google Scholar

[20]

D. Takács and G. Stépán, Experiments on quasiperiodic wheel shimmy, J. Comput. Nonlinear Dynam., 4 (2009), 031007. Google Scholar

[21]

H. K. Wang and G. Chen, Asymptotic locations of eigenfrequencies of Euler–Bernoulli beam with nonhomogeneous structural and viscous damping coefficients, SIAM J. Control Optim., 29 (1991), 347-367.  doi: 10.1137/0329019.  Google Scholar

[22]

W. Weaver, Jr., S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, 5$^{nd}$ edition, John Wiley & Sons Limited, 1990. Google Scholar

[23]

J. Wilkening and X. Zhao, Quasi-periodic traveling gravity-capillary waves, preprint, arXiv: 2002.09487. Google Scholar

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