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

Previous Article
Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents
DCDS-B Home
This Issue
Next Article
A fast high order method for time fractional diffusion equation with non-smooth data

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

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, 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.

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 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.

show all references

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.

[1]	Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[2]	Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure and Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839
[3]	Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[4]	Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[5]	Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[6]	Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
[7]	Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[8]	Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[9]	Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[10]	Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[11]	Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[12]	Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261
[13]	Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
[14]	Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[15]	Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
[16]	Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216
[17]	Michela Procesi. Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 541-552. doi: 10.3934/dcds.2005.13.541
[18]	Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9
[19]	Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[20]	Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Bochao Chen Yixian Gao
on Google Scholar
Bochao Chen Yixian Gao

2021 Impact Factor: 1.497

Tools

Article outline

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Bochao Chen Yixian Gao
on Google Scholar
Bochao Chen Yixian Gao

[Back to Top]