Abstract
In this paper, we investigate standing waves in discrete nonlinear Schrödinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bartsch, T., Liu, Z.L., Weth, T. Nodal solutions of a p-Laplacian equation. Proc. London Math. Soc, 91: 129–152 (2005)
Christodoulides, D.N., Lederer, F., Silberberg, Y. Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature, 424: 817–823 (2003)
Eilbeck, J.C., Johansson, M. The discrete nonlinear Schrodinger equation: 20 years on. In: Localization and Energy Transfer in Nonlinear Systems, edited by L. Vasquez, R.S. MacKay, M.P. Zorzano, World Scientific, Singapore, 2003, 44–67
Flach, S., Gorbach, A. Discrete breathers-advances in theory and applications. Phys. Rep., 467: 1–116 (2008)
Hardy, G.H., Littlewood, J.E., Pólya, G. Inequalities, 2nd Ed., Cambridge University Press, Cambridge, 1952
Kato, T. Perturbation theory for linear operators. Springer-Verlag, ???, 1966
Kevrekides, P.G., Rasmussen, K.Ø, Bishop, A.R. The discrete nonlinear Schroinger equation: A survey of recent results. Int. J. Modern Phys. B, 15: 2833–2900 (2001)
Kopidakis, G., Aubry, S., Tsironis, G.P. Targeted energy transfer through discrete breathers in nonlinear systems. Phys. Rev. Lett., 87: 165501 (2001)
Liu, Z.L., Su, J.B., Weth, T. Compactness results for Schroinger equations with asymptotically linear terms. J. Differential Equations, 231: 501–512 (2006)
Liu, Z.L., Sun, J.X. Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J. Differential Equations, 172: 257–299 (2001)
Livi, R., Franzosi, R., Oppo, G.L. Self-localization of Bose-Einstein condensates in optical lattices via boundary dissipation. Phys. Rev. Lett., 97: 060401 (2006)
Pankov, A. Gap solitons in periodic discrete nonlinear Schroinger equations. Nonlinearity, 19: 27–40 (2006)
Pankov, A. Gap solitons in periodic discrete nonlinear Schroinger equations, II: A generalized Nehari manifold approach. Discrete Contin. Dyn. Syst. Ser. A, 19: 419–430 (2007)
Pankov, A., Rothos, V. Periodic and decaying solutions in discrete nonlinear Schroinger equation with saturable nonlinearity. Proc. Roy. Soc. Ser. A, 464: 3219–3236 (2008)
Pankov, A. Standing waves for discrete nonlinear Schroinger equations: sign-changing nonlinearities. Appl. Anal., 92: 308–317 (2013)
Schechter, M., Zou, W. Sign-changing critical points from linking type theorems. Trans. Amer. Math. Soc., 358: 5293–5318 (2006)
Shi, H.P., Zhang, H.Q. Existence of gap solitons in periodic discrete nonlinear Schroinger equations. J. Math. Anal. Appl., 361: 411–419 (2010)
Sun, J.X. Nonlinear functional analysis and its applications. Science Press, Beijing, 2008 (in Chinese)
Teschl, G. Jacobi operators and completely integrable nonlinear lattices. Amer. Math. Soc, Providence, RI, New York, 2000
Weidmann, J. Linear operators in Hilbert spaces. Springer-Verlag, New York, 1980
Yang, M.B., Chen, W.X., Ding, Y.H. Solutions for discrete periodic Schroinger equations with spectrum 0. Acta Appl. Math., 110: 1475–1488 (2010)
Zhang, G. Breather solutions of the discrete nonlinear Schroinger equations with sign changing nonlinearity. J. Math. Phys., 52: 043516 (2011)
Zhang, G. Breather solutions of the discrete nonlinear Schroinger equations with unbounded potentials. J. Math. Phys., 50: 013505 (2009)
Zhang, G., Pankov, A. Standing waves of the discrete nonlinear Schroinger equations with unbounded potentials, II. Appl. Anal, 89: 1541–1557 (2010)
Zhou, Z., Yu, J.S. On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems. J. Differential Equations, 249: 1199–1212 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by Science and technology plan foundation of Guangzhou (No. 201607010218) and by Public Research & Capacity-Building Project of Guangdong (No. 2015A070704059).
Rights and permissions
About this article
Cite this article
He, Ts., Zhang, M., Liang, Kh. et al. Standing Waves for Discrete Nonlinear Schrödinger Equations with Nonperiodic Bounded Potentials. Acta Math. Appl. Sin. Engl. Ser. 35, 374–385 (2019). https://doi.org/10.1007/s10255-018-0787-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-018-0787-1