# American Institute of Mathematical Sciences

February  2022, 27(2): 921-944. doi: 10.3934/dcdsb.2021075

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

 1 School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, Jilin 130024, China 2 School of College of Mathematics, Jilin University, Changchun, Jilin 130012, China

* Corresponding author: Yixian Gao

Received  August 2020 Revised  January 2021 Published  February 2022 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.

Citation: Bochao Chen, Yixian Gao. Quasi-periodic travelling waves for beam equations with damping on 3-dimensional rectangular tori. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 921-944. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] R. Feola and F. Giuliani, Quasi-periodic traveling waves on an infinitely deep fluid under gravity, preprint, arXiv: 2005.08280. [12] L. Herrmann, Vibration of the Euler–Bernoulli beam with allowance for dampings, in Proc. World Congr. Eng., London, UK, (2008), 901–904. [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. [14] A. Jenkins, Self-oscillation, Phys. Rep., 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007. [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. [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. [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. [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. [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. [20] D. Takács and G. Stépán, Experiments on quasiperiodic wheel shimmy, J. Comput. Nonlinear Dynam., 4 (2009), 031007. [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. [22] W. Weaver, Jr., S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, 5$^{nd}$ edition, John Wiley & Sons Limited, 1990. [23] J. Wilkening and X. Zhao, Quasi-periodic traveling gravity-capillary waves, preprint, arXiv: 2002.09487.

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##### References:
 [1] M. Berti, L. Franzoi and A. Maspero, Traveling quasi-periodic water waves with constant vorticity, preprint, arXiv: 2004.08905. [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. [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. [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. [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. [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. [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. [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. [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. [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. [11] R. Feola and F. Giuliani, Quasi-periodic traveling waves on an infinitely deep fluid under gravity, preprint, arXiv: 2005.08280. [12] L. Herrmann, Vibration of the Euler–Bernoulli beam with allowance for dampings, in Proc. World Congr. Eng., London, UK, (2008), 901–904. [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. [14] A. Jenkins, Self-oscillation, Phys. Rep., 525 (2013), 167-222.  doi: 10.1016/j.physrep.2012.10.007. [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. [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. [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. [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. [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. [20] D. Takács and G. Stépán, Experiments on quasiperiodic wheel shimmy, J. Comput. Nonlinear Dynam., 4 (2009), 031007. [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. [22] W. Weaver, Jr., S. P. Timoshenko and D. H. Young, Vibration Problems in Engineering, 5$^{nd}$ edition, John Wiley & Sons Limited, 1990. [23] J. Wilkening and X. Zhao, Quasi-periodic traveling gravity-capillary waves, preprint, arXiv: 2002.09487.
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