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 & Continuous Dynamical Systems - A, 2018, 38 (2) : 697-714. doi: 10.3934/dcds.2018030
References:
[1]

R. AdamiC. CacciapuotiD. 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.  Google Scholar

[2]

R. AdamiC. CacciapuotiD. 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.  Google Scholar

[3]

R. AdamiE. Serra and P. Tilli, NLS ground states on graphs, Calc. Var., 54 (2015), 743-761.  doi: 10.1007/s00526-014-0804-z.  Google Scholar

[4]

R. AdamiE. 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.  Google Scholar

[5]

R. AdamiE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313 (1999), 15-37.  doi: 10.1007/s002080050248.  Google Scholar

[9]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs Amer. Math. Soc., Providence, R. I., 2013.  Google Scholar

[10]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.  Google Scholar

[11]

C. CacciapuotiD. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Kuchment, Quantum graphs, Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks Springer, Chem, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. NojaD. 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.  Google Scholar

[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.  Google Scholar

[25]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27-40.  doi: 10.1088/0951-7715/19/1/002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Pankov, Gap solitons in almost periodic one-dimensional structures, Calc. Var., 54 (2015), 1963-1984.  doi: 10.1007/s00526-015-0851-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅰ. Functional Analysis Academic Press, San Diego, 1980.  Google Scholar

[32]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators Academic Press, San Diego, 1978.  Google Scholar

[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.  Google Scholar

[34]

M. Willem, Minimax Methods Birkhäuser, Boston, 1996. Google Scholar

show all references

References:
[1]

R. AdamiC. CacciapuotiD. 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.  Google Scholar

[2]

R. AdamiC. CacciapuotiD. 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.  Google Scholar

[3]

R. AdamiE. Serra and P. Tilli, NLS ground states on graphs, Calc. Var., 54 (2015), 743-761.  doi: 10.1007/s00526-014-0804-z.  Google Scholar

[4]

R. AdamiE. 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.  Google Scholar

[5]

R. AdamiE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

T. Bartsch and Y. Ding, On a nonlinear Schrödinger equation with periodic potential, Math. Ann., 313 (1999), 15-37.  doi: 10.1007/s002080050248.  Google Scholar

[9]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs Amer. Math. Soc., Providence, R. I., 2013.  Google Scholar

[10]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.  Google Scholar

[11]

C. CacciapuotiD. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Kuchment, Quantum graphs, Ⅰ. Some basic structures, Waves Random Media, 14 (2004), S107-S128.  doi: 10.1088/0959-7174/14/1/014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks Springer, Chem, 2014. doi: 10.1007/978-3-319-04621-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. NojaD. 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.  Google Scholar

[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.  Google Scholar

[25]

A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27-40.  doi: 10.1088/0951-7715/19/1/002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

A. Pankov, Gap solitons in almost periodic one-dimensional structures, Calc. Var., 54 (2015), 1963-1984.  doi: 10.1007/s00526-015-0851-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅰ. Functional Analysis Academic Press, San Diego, 1980.  Google Scholar

[32]

M. Reed and B. Simon, Methods of Modern Mathematical Physics, Ⅳ. Analysis of Operators Academic Press, San Diego, 1978.  Google Scholar

[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.  Google Scholar

[34]

M. Willem, Minimax Methods Birkhäuser, Boston, 1996. Google Scholar

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