American Institute of Mathematical Sciences

Advanced
May  2007, 19(2): 419-430. doi: 10.3934/dcds.2007.19.419

Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach

A. Pankov 1,

1. 

Mathematics Department, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, United States

Received  May 2005 Revised  July 2005 Published  July 2007

In the present article we obtain the existence of so-called ground gap solitons in discrete periodic nonlinear Schrödinger equations with cubic nonlinearity. To do that we employ a periodic approximation technique and a generalized Nehari manifold approach.
Keywords: ground states, Nehari manifold., Gap solitons, discrete nonlinear Schrödinger equation.
Mathematics Subject Classification: 35Q55, 35Q51, 39A12, 39A70, 78A4.
Citation: A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419
[1]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[2]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[3]

Chuangye Liu, Zhi-Qiang Wang. A complete classification of ground-states for a coupled nonlinear Schrödinger system. Communications on Pure & Applied Analysis, 2017, 16 (1) : 115-130. doi: 10.3934/cpaa.2017005
[4]

Zupei Shen, Zhiqing Han, Qinqin Zhang. Ground states of nonlinear Schrödinger equations with fractional Laplacians. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2115-2125. doi: 10.3934/dcdss.2019136
[5]

Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
[6]

Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
[7]

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867
[8]

Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061
[9]

Xianhua Tang, Sitong Chen. Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4973-5002. doi: 10.3934/dcds.2017214
[10]

Sitong Chen, Junping Shi, Xianhua Tang. Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5867-5889. doi: 10.3934/dcds.2019257
[11]

Chang-Lin Xiang. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5789-5800. doi: 10.3934/dcds.2016054
[12]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
[13]

Vianney Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1961-1993. doi: 10.3934/dcds.2014.34.1961
[14]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[15]

Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091
[16]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[17]

César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067
[18]

Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117
[19]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[20]

Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020095

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (63)
  • HTML views (0)
  • Cited by (43)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences