# American Institute of Mathematical Sciences

2005, 2005(Special): 756-767. doi: 10.3934/proc.2005.2005.756

## Subharmonic bifurcations of localized solutions of a discrete NLS equation

 1 School of Mathematical Sciences, Queen Mary College, Mile End, E1 4NS London, United Kingdom

Received  September 2004 Revised  May 2005 Published  September 2005

Using an analytical approach, we derive an explicit formula for the subharmonic Mel'nikov potential ${\rm L}^{^{{\p}/{\q}}}$ for perturbations of twist maps. Our method based on the integrability of map and the variational approach of twist map. If ${\rm L}^{^{{\p}/{\q}}}$ is non--constant the perturbed twist map is non--integrable and all the resonant curves are destroyed for $\abs{\varepsilon}\ll 1$. We also apply our result to show the existence of such subharmonic bifurcations for a mapping representing localized oscillatory solutions of a discrete NLS equation with conservative and dissipative perturbations.
Citation: Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756
 [1] Kazuyuki Yagasaki. Application of the subharmonic Melnikov method to piecewise-smooth systems. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2189-2209. doi: 10.3934/dcds.2013.33.2189 [2] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047 [3] Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343 [4] Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69 [5] Jean-Pierre Eckmann, C. Eugene Wayne. Breathers as metastable states for the discrete NLS equation. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6091-6103. doi: 10.3934/dcds.2018136 [6] Scipio Cuccagna. Orbitally but not asymptotically stable ground states for the discrete NLS. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 105-134. doi: 10.3934/dcds.2010.26.105 [7] Panayotis Panayotaros. Continuation and bifurcations of breathers in a finite discrete NLS equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1227-1245. doi: 10.3934/dcdss.2011.4.1227 [8] Panayotis Panayotaros, Felipe Rivero. Multistability and localized attractors in a dissipative discrete NLS equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1137-1154. doi: 10.3934/dcdsb.2014.19.1137 [9] Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162 [10] Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255 [11] Jialin Hong, Lijun Miao, Liying Zhang. Convergence analysis of a symplectic semi-discretization for stochastic nls equation with quadratic potential. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4295-4315. doi: 10.3934/dcdsb.2019120 [12] Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258 [13] M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202 [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] Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063 [16] Anouar Bahrouni, Marek Izydorek, Joanna Janczewska. Subharmonic solutions for a class of Lagrangian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1841-1850. doi: 10.3934/dcdss.2019121 [17] Clark Robinson. Uniform subharmonic orbits for Sitnikov problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 647-652. doi: 10.3934/dcdss.2008.1.647 [18] Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995 [19] Michael C. Sullivan. Invariants of twist-wise flow equivalence. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 475-484. doi: 10.3934/dcds.1998.4.475 [20] Daniel Núñez, Pedro J. Torres. Periodic solutions of twist type of an earth satellite equation. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 303-306. doi: 10.3934/dcds.2001.7.303

Impact Factor: