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Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 8
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Articles

Orbital stability of standing waves for supercritical NLS with potential on graphs

Pages 1359-1372 | Received 29 May 2018, Accepted 27 Sep 2018, Published online: 11 Oct 2018

References

  • Berkolaiko G, Carlson C, Fulling S, et al. Quantum graphs and their applications. Providence, RI: American Mathematical Society; 2006. (Contemporary Mathematics; 415).
  • Mehmeti FA, von Below J, Nicaise S, editors. Partial differential equations on multistrucutres. New York: Marcel Dekker, Inc.; 2001. (Lecture Notes in Pure and Applied Mathematics; 219).
  • Uecker H, Grieser D, Sobirov Z, et al. Soliton transport in tabular networks: transmission at vertices in the shrinking limit. Phys Rev E. 2015;91:023209 (8 pages). doi: 10.1103/PhysRevE.91.023209
  • Adami R, Cacciapuoti C, Finco D, et al. Fast solitons on star graphs. Rev Math Phys. 2011;23(4):409–451. doi: 10.1142/S0129055X11004345
  • Adami R, Cacciapuoti C, Finco D, et al. On the structure of critical energy levels for the cubic focusing NLS on star graphs. J Phys A Math Theor. 2012;45:192001. 7pp. doi: 10.1088/1751-8113/45/19/192001
  • Adami R, Cacciapuoti C, Finco D, et al. Stationary states of NLS on star graphs. Europhys Lett. 2012;100:1003. doi: 10.1209/0295-5075/100/19901
  • Adami R, Cacciapuoti C, Finco D, et al. Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann Inst H Poincaré Anal Non Linéaire. 2014;31(6):1289–1310. doi: 10.1016/j.anihpc.2013.09.003
  • Adami R, Cacciapuoti C, Finco D, et al. Variational properties and orbital stability of standing waves for NLS equation on a star graph. J Differ Eq. 2014;257:3738–3777. doi: 10.1016/j.jde.2014.07.008
  • Adami R, Noja D. Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect. J Phys A Math Theor. 2009;42(49):495302–495320. doi: 10.1088/1751-8113/42/49/495302
  • Adami R, Noja D. Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ′ interaction. Comm Math Phys. 2013;318(1):247–289. doi: 10.1007/s00220-012-1597-6
  • Adami R, Noja D, Damanik D, et al. Exactly solvable models and bifurcations: the case of the cubic NLS with a δ or a δ′ interaction in dimension one. Math Model Nat Phenom. 2014;9(5):1–16. doi: 10.1051/mmnp/20149501
  • Adami R, Serra E, Tilli P. Threshold phenomena and existence results for NLS ground states on metric graphs. J Funct Anal. 2016;271(1):201–223. doi: 10.1016/j.jfa.2016.04.004
  • Adami R, Serra E, Tilli P. Negative energy ground states for the L2-critical NLSE on metric graphs. Commun Math Phys. 2017;352(1):387–406. doi: 10.1007/s00220-016-2797-2
  • Ardila AH. Logarithmic NLS equation on star graphs: existence and stability of standing waves. DIE. 2017;30(9/10):735–762.
  • Banica V, Ignat LI. Dispersion for the Schrödinger equation on the line with multiple dirac delta potentials and on delta trees. Anal PDE. 2014;7:903–927. doi: 10.2140/apde.2014.7.903
  • Cacciapuoti C. Existence of the ground state for the NLS with potential on graphs. 2017. arXiv:1707.07326.
  • Cacciapuoti C, Finco D, Noja D. Ground state and orbital stability for the NLS equation on a general starlike graph with potentials. Nonlinearity. 2017;30(8):3271–3303. doi: 10.1088/1361-6544/aa7cc3
  • Noja D. Nonlinear Schrödinger equation on graphs: recent results and open problems. Philos Trans R Soc Lond Ser A Math Phys Eng Sci. 2014;372:20130002 (20 pages).
  • Ardila AH. Stability of ground states for logarithmic Schrödinger equation with a δ′- interaction. Evol Equ Control Theory. 2017;6:155–175. doi: 10.3934/eect.2017009
  • Fukuizumi R, Jeanjean L. Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. Discrete Contin Dyn Syst. 2008;21:121–136. doi: 10.3934/dcds.2008.21.121
  • Fukuizumi R, Ohta M, Ozawa T. Nonlinear Schrödinger equation with a point defect. Ann Inst H Poincaré Anal Non Linéaire. 2008;25(5):837–845. doi: 10.1016/j.anihpc.2007.03.004
  • Le Coz S, Fukuizumi R, Fibich G, et al. Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Phys D. 2008;237:1103–1128. doi: 10.1016/j.physd.2007.12.004
  • Adami R, Serra E, Tilli P. Nonlinear dynamics on branched structures and networks. Riv Mat Univ Parma. 2017;8(1):109–159.
  • Cazenave T. Semilinear Schrödinger equations. Providence (RI): American Mathematical Society, Courant Institute of Mathematical Sciences; 2003. (Courant Lecture Notes in Mathematics; 10).
  • Bellazzini J, Boussaid N, Jeanjean L, et al. Existence and stability of standing waves for supercritical NLS with a partial confinement. Commun Math Phys. 2017;353(1):229–251. doi: 10.1007/s00220-017-2866-1
  • Luo X. Stability and multiplicity of standing waves for the inhomogeneous NLS equation with a harmonic potential. Nonlinear Anal Real World Appl. 2019;45:688–703. doi: 10.1016/j.nonrwa.2018.07.031
  • Noris B, Tavares H, Verzini G. Existence and orbital stability of the ground states with prescribed mass for the L2-critical and supercritical NLS on bounded domains. Anal PDE. 2014;7(8):1807–1838. doi: 10.2140/apde.2014.7.1807
  • Pierotti D, Verzinian G. Normalized bound states for the nonlinear Schrödinger equation in bounded domains. Calc Var Partial Differ Eq. 2017;56(5). Art. 133, 27 pp. doi: 10.1007/s00526-017-1232-7
  • Angulo J, Goloshchapova N. Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph. Adv Differ Eq. arXiv:1507.02312v5. 50pp; 2018.
  • Vázquez JL. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim. 1984;12:191–202. doi: 10.1007/BF01449041

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