# American Institute of Mathematical Sciences

February  2018, 38(2): 697-714. doi: 10.3934/dcds.2018030

## Nonlinear Schrödinger Equations on Periodic Metric Graphs

 1 Mathematics Department, Morgan State University, Baltimore, MD 21251, USA 2 RUDN University, Moscow 117198, Russia

Received  June 2017 Revised  August 2017 Published  February 2018

The paper is devoted to the nonlinear Schrödinger equation with periodic linear and nonlinear potentials on periodic metric graphs. Assuming that the spectrum of linear part does not contain zero, we prove the existence of finite energy ground state solution which decays exponentially fast at infinity. The proof is variational and makes use of the generalized Nehari manifold for the energy functional combined with periodic approximations. Actually, a finite energy ground state solution is obtained from periodic solutions in the infinite wave length limit.

Citation: Alexander Pankov. Nonlinear Schrödinger Equations on Periodic Metric Graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030
##### References:
 [1] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Constrained energy minimization and orbital stability for the NLS equation on a star graph, Ann. Inst. H. Poincaré, Anal. Nonlin., 31 (2014), 1289-1310.  doi: 10.1016/j.anihpc.2013.09.003. [2] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differ. Equat., 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008. [3] R. Adami, E. Serra and P. Tilli, NLS ground states on graphs, Calc. Var., 54 (2015), 743-761.  doi: 10.1007/s00526-014-0804-z. [4] R. Adami, E. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground state on graphs, J. Funct. Anal., 271 (2016), 201-223.  doi: 10.1016/j.jfa.2016.04.004. [5] R. Adami, E. Serra and P. Tilli, Negative energy ground states for the $L^2$-critical NLSE on metric graphs, Commun. Math. Phys., 352 (2017), 387-406.  doi: 10.1007/s00220-016-2797-2. [6] S. Akduman and A. Pankov, Schrödinger operators with locally integrable potentials on infinite metric graphs, Applicable Anal., 96 (2016), 2149-2161.  doi: 10.1080/00036811.2016.1207247. [7] S. Akduman and A. Pankov, Exponential decay of eigenfunctions of Schrödinger operators on infinite metric graphs, Compl. Variables Elliptic Equat., 62 (2017), 957-966.  doi: 10.1080/17476933.2016.1254204. [8] T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313 (1999), 15-37.  doi: 10.1007/s002080050248. [9] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs Amer. Math. Soc., Providence, R. I., 2013. [10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011. [11] C. Cacciapuoti, D. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general star-like graph with potentials, Nonlinearity, 30 (2017), 3271-3303. [12] P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations 2nd edition, Birkhäuser, Basel, 2013. doi: 10.1007/978-3-0348-0387-8. [13] S. Gilg, D. Pelinovsky and G. Schneider, Validity of the NLS approximation for periodic quantum graphs Nonlinear Differ. Equ. Appl. 23 (2016), Art. 63, 30 pp. doi: 10.1007/s00030-016-0417-7. [14] E. Korotyaev and L. Lobanov, Schrödinger operators on zigzag nanotubes, Ann. Inst. H. Poincaré, 8 (2007), 1151-1176.  doi: 10.1007/s00023-007-0331-y. [15] P. Kuchment, Graph models for waves in thin structures, Waves Random Media, 12 (2002), R1-R24.  doi: 10.1088/0959-7174/12/4/201. [16] P. Kuchment, Quantum graphs, Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014. [17] P. Kuchment, Quantum graphs: Ⅱ. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen., 38 (2005), 4887-4900.  doi: 10.1088/0305-4470/38/22/013. [18] P. Kuchment and O. Post, On the spectra of carbon nano-structures, Commun. Math. Phys., 275 (2007), 805-826.  doi: 10.1007/s00220-007-0316-1. [19] J. L. Marzuola and D. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express, 2016 (2016), 98-145.  doi: 10.1093/amrx/abv011. [20] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks Springer, Chem, 2014. doi: 10.1007/978-3-319-04621-1. [21] H. Niikuni, Decisiveness of the spectral gaps of periodic periodic Schrödinger operators on the dumbbell-like metric graph, Opusc. Math., 35 (2015), 199-234.  doi: 10.7494/OpMath.2015.35.2.199. [22] D. Noja, Nonlinear Schrödinger equation on graphs: Recent results and open problems Phil. Trans. Roy. Soc. 372 (2014), 20130002, 20pp. doi: 10.1098/rsta.2013.0002. [23] D. Noja, D. Pelinovsky and G. Shaikhova, Bifurcation and stability of standing waves on tadpole graphs, Nonlinearity, 28 (2015), 2343-2378.  doi: 10.1088/0951-7715/28/7/2343. [24] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8. [25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27-40.  doi: 10.1088/0951-7715/19/1/002. [26] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations Ⅱ: A generalized Nehari manifold approach, Discr. Cont. Dyn. Syst., 19 (2007), 419-430.  doi: 10.3934/dcds.2007.19.419. [27] A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.  doi: 10.1090/S0002-9939-08-09484-7. [28] A. Pankov, Gap solitons in almost periodic one-dimensional structures, Calc. Var., 54 (2015), 1963-1984.  doi: 10.1007/s00526-015-0851-0. [29] A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discr. Cont. Dyn. Syst., 30 (2011), 835-849.  doi: 10.3934/dcds.2011.30.835. [30] D. Pelinovsky and G. Schneider, Bifurcation of standing localized waves on periodic graphs, Ann. H. Poincaré, 18 (2017), 1185-1211.  doi: 10.1007/s00023-016-0536-z. [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅰ. Functional Analysis Academic Press, San Diego, 1980. [32] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators Academic Press, San Diego, 1978. [33] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal, 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013. [34] M. Willem, Minimax Methods Birkhäuser, Boston, 1996.

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##### References:
 [1] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Constrained energy minimization and orbital stability for the NLS equation on a star graph, Ann. Inst. H. Poincaré, Anal. Nonlin., 31 (2014), 1289-1310.  doi: 10.1016/j.anihpc.2013.09.003. [2] R. Adami, C. Cacciapuoti, D. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differ. Equat., 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008. [3] R. Adami, E. Serra and P. Tilli, NLS ground states on graphs, Calc. Var., 54 (2015), 743-761.  doi: 10.1007/s00526-014-0804-z. [4] R. Adami, E. Serra and P. Tilli, Threshold phenomena and existence results for NLS ground state on graphs, J. Funct. Anal., 271 (2016), 201-223.  doi: 10.1016/j.jfa.2016.04.004. [5] R. Adami, E. Serra and P. Tilli, Negative energy ground states for the $L^2$-critical NLSE on metric graphs, Commun. Math. Phys., 352 (2017), 387-406.  doi: 10.1007/s00220-016-2797-2. [6] S. Akduman and A. Pankov, Schrödinger operators with locally integrable potentials on infinite metric graphs, Applicable Anal., 96 (2016), 2149-2161.  doi: 10.1080/00036811.2016.1207247. [7] S. Akduman and A. Pankov, Exponential decay of eigenfunctions of Schrödinger operators on infinite metric graphs, Compl. Variables Elliptic Equat., 62 (2017), 957-966.  doi: 10.1080/17476933.2016.1254204. [8] T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313 (1999), 15-37.  doi: 10.1007/s002080050248. [9] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs Amer. Math. Soc., Providence, R. I., 2013. [10] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011. [11] C. Cacciapuoti, D. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general star-like graph with potentials, Nonlinearity, 30 (2017), 3271-3303. [12] P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations 2nd edition, Birkhäuser, Basel, 2013. doi: 10.1007/978-3-0348-0387-8. [13] S. Gilg, D. Pelinovsky and G. Schneider, Validity of the NLS approximation for periodic quantum graphs Nonlinear Differ. Equ. Appl. 23 (2016), Art. 63, 30 pp. doi: 10.1007/s00030-016-0417-7. [14] E. Korotyaev and L. Lobanov, Schrödinger operators on zigzag nanotubes, Ann. Inst. H. Poincaré, 8 (2007), 1151-1176.  doi: 10.1007/s00023-007-0331-y. [15] P. Kuchment, Graph models for waves in thin structures, Waves Random Media, 12 (2002), R1-R24.  doi: 10.1088/0959-7174/12/4/201. [16] P. Kuchment, Quantum graphs, Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014. [17] P. Kuchment, Quantum graphs: Ⅱ. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen., 38 (2005), 4887-4900.  doi: 10.1088/0305-4470/38/22/013. [18] P. Kuchment and O. Post, On the spectra of carbon nano-structures, Commun. Math. Phys., 275 (2007), 805-826.  doi: 10.1007/s00220-007-0316-1. [19] J. L. Marzuola and D. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express, 2016 (2016), 98-145.  doi: 10.1093/amrx/abv011. [20] D. Mugnolo, Semigroup Methods for Evolution Equations on Networks Springer, Chem, 2014. doi: 10.1007/978-3-319-04621-1. [21] H. Niikuni, Decisiveness of the spectral gaps of periodic periodic Schrödinger operators on the dumbbell-like metric graph, Opusc. Math., 35 (2015), 199-234.  doi: 10.7494/OpMath.2015.35.2.199. [22] D. Noja, Nonlinear Schrödinger equation on graphs: Recent results and open problems Phil. Trans. Roy. Soc. 372 (2014), 20130002, 20pp. doi: 10.1098/rsta.2013.0002. [23] D. Noja, D. Pelinovsky and G. Shaikhova, Bifurcation and stability of standing waves on tadpole graphs, Nonlinearity, 28 (2015), 2343-2378.  doi: 10.1088/0951-7715/28/7/2343. [24] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8. [25] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27-40.  doi: 10.1088/0951-7715/19/1/002. [26] A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations Ⅱ: A generalized Nehari manifold approach, Discr. Cont. Dyn. Syst., 19 (2007), 419-430.  doi: 10.3934/dcds.2007.19.419. [27] A. Pankov, On decay of solutions to nonlinear Schrödinger equations, Proc. Amer. Math. Soc., 136 (2008), 2565-2570.  doi: 10.1090/S0002-9939-08-09484-7. [28] A. Pankov, Gap solitons in almost periodic one-dimensional structures, Calc. Var., 54 (2015), 1963-1984.  doi: 10.1007/s00526-015-0851-0. [29] A. Pankov and V. Rothos, Traveling waves in Fermi-Pasta-Ulam lattices with saturable nonlinearities, Discr. Cont. Dyn. Syst., 30 (2011), 835-849.  doi: 10.3934/dcds.2011.30.835. [30] D. Pelinovsky and G. Schneider, Bifurcation of standing localized waves on periodic graphs, Ann. H. Poincaré, 18 (2017), 1185-1211.  doi: 10.1007/s00023-016-0536-z. [31] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅰ. Functional Analysis Academic Press, San Diego, 1980. [32] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators Academic Press, San Diego, 1978. [33] A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal, 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013. [34] M. Willem, Minimax Methods Birkhäuser, Boston, 1996.
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