Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

Previous Article
Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$
CPAA Home
This Issue
Next Article
Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

September 2015, 14(5): 1929-1940. doi: 10.3934/cpaa.2015.14.1929

Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

1.	College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

Received November 2014 Revised January 2015 Published June 2015

Abstract
Related Papers

Based on a generalized linking theorem for the strongly indefinite functionals, we study the existence of homoclinic orbits of the second order self-adjoint discrete Hamiltonian system \begin{eqnarray} \triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n))=0, \end{eqnarray} where $p(n), L(n)$ and $W(n, x)$ are $N$-periodic on $n$, and $0$ lies in a gap of the spectrum $\sigma(\mathcal{A})$ of the operator $\mathcal{A}$, which is bounded self-adjoint in $l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$ defined by $(\mathcal{A}u)(n)=\triangle [p(n)\triangle u(n-1)]-L(n)u(n)$. Under weak superquadratic conditions, we establish the existence of homoclinic orbits.

Keywords: Discrete Hamiltonian system, homoclinic orbit, strongly indefinite functional., periodicity.

Mathematics Subject Classification: Primary: 39A11; Secondary: 58E05, 70H0.

Citation: Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

References:

[1]	R. P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications,, second edition, (2000).
[2]	C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations,, Kluwer Texts in the Mathematical Sciences, (1996). doi: 10.1007/978-1-4757-2467-7.
[3]	C. J. Batkam, Homoclinic orbits of first-order superquadratic Hamiltonian systems,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 3353. doi: 10.3934/dcds.2014.34.3353.
[4]	J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger Hamiltonian with potentials of order zero,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 1061. doi: 10.3934/dcds.2013.33.1061.
[5]	X. Q. Deng and G. Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign,, \emph{Acta Appl. Math.}, 103 (2008), 301. doi: 10.1007/s10440-008-9237-z.
[6]	D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators,, Oxford Mathematical Monographs, (1987).
[7]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems,, \emph{J. Differential Equations}, 219 (2005), 375. doi: 10.1016/j.jde.2005.06.029.
[8]	G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part,, \emph{Commun. Contemp. Math.}, 4 (2002), 763. doi: 10.1142/S0219199702000853.
[9]	X. Y. Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems,, \emph{J. Math. Anal. Appl.}, 373 (2011), 59. doi: 10.1016/j.jmaa.2010.06.008.
[10]	X. Y. Lin and X. H. Tang, Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential,, \emph{Advances in Difference Equations}, 154 (2013).
[11]	M. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations,, \emph{Nonlinear Anal.}, 67 (2007), 1737. doi: 10.1016/j.na.2006.08.014.
[12]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, \emph{Differential Integral Equations}, 5 (1992), 1115.
[13]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, \emph{Proc, 114 (1990), 33. doi: 10.1017/S0308210500024240.
[14]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, \emph{Math. Z.}, 206 (1991), 473. doi: 10.1007/BF02571356.
[15]	J. Sun, J. Chu and Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum zero,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 3807. doi: 10.3934/dcds.2013.33.3807.
[16]	X. H. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems,, \emph{Advances in Difference Equations}, 242 (2013). doi: 10.1186/1687-1847-2013-242.
[17]	X. H. Tang and X. Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential,, \emph{J. Differ. Equ. Appl.}, 17 (2011), 1617. doi: 10.1080/10236191003730514.
[18]	X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential,, \emph{J. Differ. Equ. Appl.}, 19 (2013), 796. doi: 10.1080/10236198.2012.691168.
[19]	X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation,, \emph{J. Aust. Math. Soc.}, 98 (2015), 104. doi: 10.1017/S144678871400041X.
[20]	J. S. Yu, H. Shi and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation,, \emph{J. Math. Anal. Appl.}, 352 (2009), 799. doi: 10.1016/j.jmaa.2008.11.043.
[21]	Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems,, \emph{Proc. Amer. Math. Soc.}, ().
[22]	V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems,, \emph{Math. Ann.}, 288 (1990), 133. doi: 10.1007/BF01444526.
[23]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials,, \emph{J. Amer. Math. Soc.}, 4 (1991), 693. doi: 10.2307/2939286.
[24]	Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems,, \emph{J. Differential Equations}, 249 (2010), 1199. doi: 10.1016/j.jde.2010.03.010.
[25]	Z. Zhou and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity,, \emph{Acta Mathematica Sinica}, 29 (2013), 1809. doi: 10.1007/s10114-013-0736-0.
[26]	Z. Zhou, J. S. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity,, \emph{Science China, 54 (2011), 83. doi: 10.1007/s11425-010-4101-9.

show all references

References:

[1]	R. P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications,, second edition, (2000).
[2]	C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations,, Kluwer Texts in the Mathematical Sciences, (1996). doi: 10.1007/978-1-4757-2467-7.
[3]	C. J. Batkam, Homoclinic orbits of first-order superquadratic Hamiltonian systems,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 3353. doi: 10.3934/dcds.2014.34.3353.
[4]	J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger Hamiltonian with potentials of order zero,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 1061. doi: 10.3934/dcds.2013.33.1061.
[5]	X. Q. Deng and G. Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign,, \emph{Acta Appl. Math.}, 103 (2008), 301. doi: 10.1007/s10440-008-9237-z.
[6]	D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators,, Oxford Mathematical Monographs, (1987).
[7]	M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems,, \emph{J. Differential Equations}, 219 (2005), 375. doi: 10.1016/j.jde.2005.06.029.
[8]	G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part,, \emph{Commun. Contemp. Math.}, 4 (2002), 763. doi: 10.1142/S0219199702000853.
[9]	X. Y. Lin and X. H. Tang, Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems,, \emph{J. Math. Anal. Appl.}, 373 (2011), 59. doi: 10.1016/j.jmaa.2010.06.008.
[10]	X. Y. Lin and X. H. Tang, Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential,, \emph{Advances in Difference Equations}, 154 (2013).
[11]	M. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations,, \emph{Nonlinear Anal.}, 67 (2007), 1737. doi: 10.1016/j.na.2006.08.014.
[12]	W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, \emph{Differential Integral Equations}, 5 (1992), 1115.
[13]	P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, \emph{Proc, 114 (1990), 33. doi: 10.1017/S0308210500024240.
[14]	P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, \emph{Math. Z.}, 206 (1991), 473. doi: 10.1007/BF02571356.
[15]	J. Sun, J. Chu and Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum zero,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013), 3807. doi: 10.3934/dcds.2013.33.3807.
[16]	X. H. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems,, \emph{Advances in Difference Equations}, 242 (2013). doi: 10.1186/1687-1847-2013-242.
[17]	X. H. Tang and X. Y. Lin, Existence and multiplicity of homoclinic solutions for second-order discrete Hamiltonian systems with subquadratic potential,, \emph{J. Differ. Equ. Appl.}, 17 (2011), 1617. doi: 10.1080/10236191003730514.
[18]	X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential,, \emph{J. Differ. Equ. Appl.}, 19 (2013), 796. doi: 10.1080/10236198.2012.691168.
[19]	X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation,, \emph{J. Aust. Math. Soc.}, 98 (2015), 104. doi: 10.1017/S144678871400041X.
[20]	J. S. Yu, H. Shi and Z. M. Guo, Homoclinic orbits for nonlinear difference equations containing both advance and retardation,, \emph{J. Math. Anal. Appl.}, 352 (2009), 799. doi: 10.1016/j.jmaa.2008.11.043.
[21]	Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems,, \emph{Proc. Amer. Math. Soc.}, ().
[22]	V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems,, \emph{Math. Ann.}, 288 (1990), 133. doi: 10.1007/BF01444526.
[23]	V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials,, \emph{J. Amer. Math. Soc.}, 4 (1991), 693. doi: 10.2307/2939286.
[24]	Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems,, \emph{J. Differential Equations}, 249 (2010), 1199. doi: 10.1016/j.jde.2010.03.010.
[25]	Z. Zhou and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity,, \emph{Acta Mathematica Sinica}, 29 (2013), 1809. doi: 10.1007/s10114-013-0736-0.
[26]	Z. Zhou, J. S. Yu and Y. Chen, Homoclinic solutions in periodic difference equations with saturable nonlinearity,, \emph{Science China, 54 (2011), 83. doi: 10.1007/s11425-010-4101-9.

[1]	Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[2]	Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[3]	Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021
[4]	Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139
[5]	Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107
[6]	Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[7]	W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[8]	S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493
[9]	Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128
[10]	B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217
[11]	Sergey Gonchenko, Ivan Ovsyannikov. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 273-288. doi: 10.3934/dcdss.2017013
[12]	Jacobo Pejsachowicz, Robert Skiba. Topology and homoclinic trajectories of discrete dynamical systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1077-1094. doi: 10.3934/dcdss.2013.6.1077
[13]	Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778
[14]	Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[15]	Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269
[16]	Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353
[17]	Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 719-734. doi: 10.3934/dcdsb.2005.5.719
[18]	V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[19]	Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789
[20]	Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences