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

2. 

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

* Corresponding author: Yixian Gao

Received  August 2020 Revised  January 2021 Published  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 & 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. CampbellJ. BélairT. 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. CampbellJ. BélairT. 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. DengP. 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. CampbellJ. BélairT. 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. CampbellJ. BélairT. 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. DengP. 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

[1]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[2]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241

[3]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

[4]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047

[5]

Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867

[6]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[8]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[9]

Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021060

[10]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[11]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995

[12]

Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021103

[13]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[14]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104

[15]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[16]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[17]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[18]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[19]

Lin Yang, Yejuan Wang, Tomás Caraballo. Regularity of global attractors and exponential attractors for $ 2 $D quasi-geostrophic equations with fractional dissipation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021093

[20]

Sishu Shankar Muni, Robert I. McLachlan, David J. W. Simpson. Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3629-3650. doi: 10.3934/dcds.2021010

2019 Impact Factor: 1.27

Article outline

[Back to Top]