October  2018, 38(10): 5039-5066. doi: 10.3934/dcds.2018221

On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph

Rua do Matão, 1010, Cidade Universitária, São Paulo - SP, 05508-090, Brazil

* Corresponding author: nataliia@ime.usp.br

Received  November 2017 Revised  June 2018 Published  July 2018

Fund Project: J. Angulo was supported partially by Grant CNPq/Brazil. N. Goloshchapova was supported by FAPESP under the project 2016/02060-9.

We study the nonlinear Schrödinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate the orbital instability of the standing waves $e^{i\omega t}{\bf \Phi}(x)$ of the NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile ${\bf \Phi}(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory, avoiding the variational techniques standard in the stability study. We also prove the orbital stability of the unique standing wave solution to the NLS-$\delta$ equation with repulsive nonlinearity.

Citation: Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
References:
[1]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.  doi: 10.1016/j.jde.2016.01.029.

[2]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008.

[3]

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. Non Linéaire, 31 (2014), 1289-1310.  doi: 10.1016/j.anihpc.2013.09.003.

[4]

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edition, AMS Chelsea Publishing, Providence, RI, 2005.

[5]

J. Angulo and N. Goloshchapova, Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, preprint, arXiv:1507.02312v5.

[6]

J. AnguloO. Lopes and A. Neves, Instability of travelling waves for weakly coupled KdV systems, Nonlinear Anal., 69 (2008), 1870-1887.  doi: 10.1016/j.na.2007.07.039.

[7]

J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874. 

[8]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013.

[9]

J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.

[10]

C. CacciapuotiD. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303.  doi: 10.1088/1361-6544/aa7cc3.

[11]

T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003. doi: 10.1090/cln/010.

[12]

P. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, IMRN, 24 (2011), 5505-5624.  doi: 10.1007/s11005-010-0458-5.

[13]

P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev, Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, 77, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/pspum/077.

[14]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[15]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.

[16]

D. B. HenryJ. F. Wreszinski and W. F. Perez, Stability theory for solitary-wave solutions of scalar field equations, Comm. Math. Phys., 85 (1982), 351-361.  doi: 10.1007/BF01208719.

[17]

A. Kairzhan, Orbital instability of standing waves for NLS equation on star graphs, preprint, arXiv: 1712.02773v2.

[18]

A. Kairzhan and D. E. Pelinovsky, Spectral stability of shifted states on star graphs, J. Phys. A, 51 (2018), 095203, 23 pp.

[19]

A. Kairzhan and D. E. Pelinovsky, Nonlinear instability of half-solitons on star graphs, J. Differential Equations, 264 (2018), 7357-7383.  doi: 10.1016/j.jde.2018.02.020.

[20]

M. Ohta and M. Kaminaga, Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48. 

[21]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.

[22]

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

[23]

S. Le CozR. FukuizumiG. FibichB. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D, 237 (2008), 1103-1128.  doi: 10.1016/j.physd.2007.12.004.

[24]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition, Universitext, Springer, New York, 2009.

[25]

D. Mugnolo (editor), Mathematical Technology of Networks, Bielefeld, December 2013, Springer Proceedings in Mathematics & Statistics 128, 2015. doi: 10.1007/978-3-319-16619-3.

[26]

M. A. Naimark, Linear Differential Operators, (Russian), 2nd edition, revised and augmented, Izdat. "Nauka", Moscow, 1969.

[27]

D. Noja, Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130002, 20 pp. doi: 10.1098/rsta.2013.0002.

[28]

M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., 261 (2011), 90-110.  doi: 10.1016/j.jfa.2011.03.010.

[29]

O. Post, Spectral Analysis on Graph-Like Spaces, Lecture Notes in Mathematics, 2039, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23840-6.

[30]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.

show all references

References:
[1]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.  doi: 10.1016/j.jde.2016.01.029.

[2]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008.

[3]

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. Non Linéaire, 31 (2014), 1289-1310.  doi: 10.1016/j.anihpc.2013.09.003.

[4]

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edition, AMS Chelsea Publishing, Providence, RI, 2005.

[5]

J. Angulo and N. Goloshchapova, Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, preprint, arXiv:1507.02312v5.

[6]

J. AnguloO. Lopes and A. Neves, Instability of travelling waves for weakly coupled KdV systems, Nonlinear Anal., 69 (2008), 1870-1887.  doi: 10.1016/j.na.2007.07.039.

[7]

J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874. 

[8]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013.

[9]

J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.

[10]

C. CacciapuotiD. Finco and D. Noja, Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, 30 (2017), 3271-3303.  doi: 10.1088/1361-6544/aa7cc3.

[11]

T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003. doi: 10.1090/cln/010.

[12]

P. Deift and J. Park, Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, IMRN, 24 (2011), 5505-5624.  doi: 10.1007/s11005-010-0458-5.

[13]

P. Exner, J. P. Keating, P. Kuchment, T. Sunada and A. Teplyaev, Analysis on Graphs and Its Applications, Proceedings of Symposia in Pure Mathematics, 77, American Mathematical Society, Providence, RI, 2008. doi: 10.1090/pspum/077.

[14]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.

[15]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.

[16]

D. B. HenryJ. F. Wreszinski and W. F. Perez, Stability theory for solitary-wave solutions of scalar field equations, Comm. Math. Phys., 85 (1982), 351-361.  doi: 10.1007/BF01208719.

[17]

A. Kairzhan, Orbital instability of standing waves for NLS equation on star graphs, preprint, arXiv: 1712.02773v2.

[18]

A. Kairzhan and D. E. Pelinovsky, Spectral stability of shifted states on star graphs, J. Phys. A, 51 (2018), 095203, 23 pp.

[19]

A. Kairzhan and D. E. Pelinovsky, Nonlinear instability of half-solitons on star graphs, J. Differential Equations, 264 (2018), 7357-7383.  doi: 10.1016/j.jde.2018.02.020.

[20]

M. Ohta and M. Kaminaga, Stability of standing waves for nonlinear Schrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48. 

[21]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.

[22]

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

[23]

S. Le CozR. FukuizumiG. FibichB. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D, 237 (2008), 1103-1128.  doi: 10.1016/j.physd.2007.12.004.

[24]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition, Universitext, Springer, New York, 2009.

[25]

D. Mugnolo (editor), Mathematical Technology of Networks, Bielefeld, December 2013, Springer Proceedings in Mathematics & Statistics 128, 2015. doi: 10.1007/978-3-319-16619-3.

[26]

M. A. Naimark, Linear Differential Operators, (Russian), 2nd edition, revised and augmented, Izdat. "Nauka", Moscow, 1969.

[27]

D. Noja, Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130002, 20 pp. doi: 10.1098/rsta.2013.0002.

[28]

M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., 261 (2011), 90-110.  doi: 10.1016/j.jfa.2011.03.010.

[29]

O. Post, Spectral Analysis on Graph-Like Spaces, Lecture Notes in Mathematics, 2039, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23840-6.

[30]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.

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