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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 |
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.
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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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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