# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
##### References:
 [1] R. Adami, C. Cacciapuoti, D. 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.  Google Scholar [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. Differential Equations, 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008.  Google Scholar [3] 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. Non Linéaire, 31 (2014), 1289-1310.  doi: 10.1016/j.anihpc.2013.09.003.  Google Scholar [4] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edition, AMS Chelsea Publishing, Providence, RI, 2005.  Google Scholar [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. Google Scholar [6] J. Angulo, O. 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.  Google Scholar [7] J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874.   Google Scholar [8] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013.  Google Scholar [9] J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.  Google Scholar [10] C. Cacciapuoti, D. 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.  Google Scholar [11] T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003. doi: 10.1090/cln/010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. Grillakis, J. 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.  Google Scholar [15] M. Grillakis, J. 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.  Google Scholar [16] D. B. Henry, J. 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.  Google Scholar [17] A. Kairzhan, Orbital instability of standing waves for NLS equation on star graphs, preprint, arXiv: 1712.02773v2. Google Scholar [18] A. Kairzhan and D. E. Pelinovsky, Spectral stability of shifted states on star graphs, J. Phys. A, 51 (2018), 095203, 23 pp.  Google Scholar [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.  Google Scholar [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.   Google Scholar [21] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [22] 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 [23] S. Le Coz, R. Fukuizumi, G. Fibich, B. 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.  Google Scholar [24] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition, Universitext, Springer, New York, 2009.  Google Scholar [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.  Google Scholar [26] M. A. Naimark, Linear Differential Operators, (Russian), 2nd edition, revised and augmented, Izdat. "Nauka", Moscow, 1969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.  Google Scholar

show all references

##### References:
 [1] R. Adami, C. Cacciapuoti, D. 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.  Google Scholar [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. Differential Equations, 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008.  Google Scholar [3] 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. Non Linéaire, 31 (2014), 1289-1310.  doi: 10.1016/j.anihpc.2013.09.003.  Google Scholar [4] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics, 2nd edition, AMS Chelsea Publishing, Providence, RI, 2005.  Google Scholar [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. Google Scholar [6] J. Angulo, O. 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.  Google Scholar [7] J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Differential Integral Equations, 29 (2016), 837-874.   Google Scholar [8] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013.  Google Scholar [9] J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.  Google Scholar [10] C. Cacciapuoti, D. 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.  Google Scholar [11] T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, AMS. Lecture Notes, v. 10, 2003. doi: 10.1090/cln/010.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] M. Grillakis, J. 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.  Google Scholar [15] M. Grillakis, J. 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.  Google Scholar [16] D. B. Henry, J. 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.  Google Scholar [17] A. Kairzhan, Orbital instability of standing waves for NLS equation on star graphs, preprint, arXiv: 1712.02773v2. Google Scholar [18] A. Kairzhan and D. E. Pelinovsky, Spectral stability of shifted states on star graphs, J. Phys. A, 51 (2018), 095203, 23 pp.  Google Scholar [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.  Google Scholar [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.   Google Scholar [21] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [22] 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 [23] S. Le Coz, R. Fukuizumi, G. Fibich, B. 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.  Google Scholar [24] F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, 2nd edition, Universitext, Springer, New York, 2009.  Google Scholar [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.  Google Scholar [26] M. A. Naimark, Linear Differential Operators, (Russian), 2nd edition, revised and augmented, Izdat. "Nauka", Moscow, 1969.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [30] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York, 1978.  Google Scholar
 [1] Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843 [2] Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525 [3] François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229 [4] Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113 [5] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031 [6] François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137 [7] Soohyun Bae, Jaeyoung Byeon. Standing waves of nonlinear Schrödinger equations with optimal conditions for potential and nonlinearity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 831-850. doi: 10.3934/cpaa.2013.12.831 [8] Hiroaki Kikuchi. Remarks on the orbital instability of standing waves for the wave-Schrödinger system in higher dimensions. Communications on Pure & Applied Analysis, 2010, 9 (2) : 351-364. doi: 10.3934/cpaa.2010.9.351 [9] J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445 [10] Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2093-2116. doi: 10.3934/cpaa.2019094 [11] Alex H. Ardila. Stability of standing waves for a nonlinear SchrÖdinger equation under an external magnetic field. Communications on Pure & Applied Analysis, 2018, 17 (1) : 163-175. doi: 10.3934/cpaa.2018010 [12] Reika Fukuizumi, Louis Jeanjean. Stability of standing waves for a nonlinear Schrödinger equation wdelta potentialith a repulsive Dirac. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 121-136. doi: 10.3934/dcds.2008.21.121 [13] Sevdzhan Hakkaev. Orbital stability of solitary waves of the Schrödinger-Boussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1043-1050. doi: 10.3934/cpaa.2007.6.1043 [14] Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825 [15] Ademir Pastor. On three-wave interaction Schrödinger systems with quadratic nonlinearities: Global well-posedness and standing waves. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2217-2242. doi: 10.3934/cpaa.2019100 [16] Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193 [17] Mourad Bellassoued, Ibtissem Ben Aïcha, Zouhour Rezig. Stable determination of a vector field in a non-Self-Adjoint dynamical Schrödinger equation on Riemannian manifolds. Mathematical Control & Related Fields, 2021, 11 (2) : 403-431. doi: 10.3934/mcrf.2020042 [18] Jaeyoung Byeon, Louis Jeanjean. Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 255-269. doi: 10.3934/dcds.2007.19.255 [19] Yonggeun Cho, Hichem Hajaiej, Gyeongha Hwang, Tohru Ozawa. On the orbital stability of fractional Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1267-1282. doi: 10.3934/cpaa.2014.13.1267 [20] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

2019 Impact Factor: 1.338

## Metrics

• PDF downloads (179)
• HTML views (128)
• Cited by (7)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]