Breathers of discrete one-dimensional nonlinear Schrödinger equations in inhomogeneous media. (English) Zbl 1420.35362

Summary: This paper is concerned with the breathers of discrete one-dimensional nonlinear Schrödinger equations in inhomogeneous media. By using a constrained minimization approach known as the Nehari variational principle or the Nehari manifold approach, we obtain the existence of nontrivial breathers.


35Q55 NLS equations (nonlinear Schrödinger equations)
37K60 Lattice dynamics; integrable lattice equations
35C08 Soliton solutions
49Q99 Manifolds and measure-geometric topics
Full Text: DOI Euclid


[1] V. Banica and L. I. Ignat, Dispersion for the Schrödinger equation on networks, J. Math. Phys. 52 (2011), no. 8, 083703, 14 pp. · Zbl 1272.81079
[2] T. Brugarino and M. Sciacca, Integrability of an inhomogeneous nonlinear Schrödinger equation in Bose-Einstein condensates and fiber optics, J. Math. Phys. 51 (2010), no. 9, 093503, 18 pp. · Zbl 1309.35127
[3] X. Chen, J.-S. Guo and C.-C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal. 189 (2008), no. 2, 189–236. · Zbl 1152.37033
[4] J. B. Conway, A Course in Functional Analysis, Graduate Texts in Mathematics 96, Springer-Verlag, New York, 1985. · Zbl 0558.46001
[5] H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, Discrete spatial optical solitons in waveguide arrays, Phys. Rev. Lett. 81 (1998), no. 16, 3383–3386.
[6] P. Gérard, Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem, International Congress of Mathematicians, Vol. III, 157–182, Eur. Math. Soc., Zürich, 2006.
[7] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, J. Funct. Anal. 32 (1979), no. 1, 1–71.
[8] N. I. Karachalios, A remark on the existence of breather solutions for the discrete nonlinear Schrödinger equation in infinite lattices: the case of site-dependent anharmonic parameters, Proc. Edinb. Math. Soc. (2) 49 (2006), no. 1, 115–129. · Zbl 1134.37363
[9] N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations 217 (2005), no. 1, 88–123.
[10] J. Li and F. Chen, Breather solutions of a generalized nonlinear Schrödinger system, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 25 (2015), no. 8, 1550105, 16 pp. · Zbl 1321.35008
[11] T. Pertsch, U. Peschel and F. Lederer, Discrete solitons in inhomogeneous waveguide arrays, Chaos 13 (2003), no. 2, 744–753.
[12] C. Sulem and P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-focusing and wave collapse, Applied Mathematical Sciences 139, Springer-Verlag, New York, 1999.
[13] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical Surveys and Monographs 72, American Mathematical Society, Providence, RI, 2000. · Zbl 1056.39029
[14] G. Zhang, Breather solutions of the discrete nonlinear Schrödinger equations with unbounded potentials, J. Math. Phys. 50 (2009), no. 1, 013505, 12 pp. · Zbl 1200.37072
[15] ——–, Breather solutions of the discrete nonlinear Schrödinger equations with sign changing nonlinearity, J. Math. Phys. 52 (2011), no. 4, 043516, 11 pp. · Zbl 1316.35270
[16] Z. Zhou and D. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Sci. China Math. 58 (2015), no. 4, 781–790. · Zbl 1328.39011
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