August  2006, 15(3): 883-903. doi: 10.3934/dcds.2006.15.883

Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$

1. 

Sissa, via Beirut 2-4, 34014 Trieste, Italy

Received  January 2005 Revised  November 2005 Published  April 2006

We consider 1D completely resonant nonlinear wave equations of the type $v_{t t}$ - $v_{x x}$$ = -v^3 + \mathcal{O}(v^4)$ with spatial periodic boundary conditions. We prove the existence of a new type of quasi-periodic small amplitude solutions with two frequencies, for more general nonlinearities. These solutions turn out to be, at the first order, the superposition of a traveling wave and a modulation of long period, depending only on time.
Citation: Pietro Baldi. Quasi-periodic solutions of the equation $v_{t t} - v_{x x} +v^3 = f(v)$. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 883-903. doi: 10.3934/dcds.2006.15.883
[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[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]

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

[4]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[5]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615

[6]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101

[7]

Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623

[8]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467

[9]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537

[10]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104

[11]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585

[12]

Chengming Cao, Xiaoping Yuan. Quasi-periodic solutions for perturbed generalized nonlinear vibrating string equation with singularities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1867-1901. doi: 10.3934/dcds.2017079

[13]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171

[14]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019

[15]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839

[16]

Jinjing Jiao, Guanghua Shi. Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5157-5180. doi: 10.3934/cpaa.2020231

[17]

Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080

[18]

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

[19]

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

[20]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]