Constrained energy minimization and orbital stability for the NLS equation on a star graph
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1289-1310.

On a star graph 𝒢, we consider a nonlinear Schrödinger equation with focusing nonlinearity of power type and an attractive Dirac's delta potential located at the vertex. The equation can be formally written as i t Ψ(t)=-ΔΨ(t)-|Ψ(t)| 2μ Ψ(t)+αδ 0 Ψ(t), where the strength α of the vertex interaction is negative and the wave function Ψ is supposed to be continuous at the vertex. The values of the mass and energy functionals are conserved by the flow. We show that for 0<μ2 the energy at fixed mass is bounded from below and that for every mass m below a critical mass m it attains its minimum value at a certain Ψ ˆ m H 1 (𝒢).Moreover, the set of minimizers has the structure ={e iθ Ψ ˆ m ,θ}. Correspondingly, for every m<m there exists a unique ω=ω(m) such that the standing wave Ψ ˆ ω e iωt is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is nontrivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to α=0.

@article{AIHPC_2014__31_6_1289_0,
     author = {Adami, Riccardo and Cacciapuoti, Claudio and Finco, Domenico and Noja, Diego},
     title = {Constrained energy minimization and orbital stability for the {NLS} equation on a star graph},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1289--1310},
     publisher = {Elsevier},
     volume = {31},
     number = {6},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.09.003},
     mrnumber = {3280068},
     zbl = {1304.81087},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.003/}
}
TY  - JOUR
AU  - Adami, Riccardo
AU  - Cacciapuoti, Claudio
AU  - Finco, Domenico
AU  - Noja, Diego
TI  - Constrained energy minimization and orbital stability for the NLS equation on a star graph
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2014
SP  - 1289
EP  - 1310
VL  - 31
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.003/
DO  - 10.1016/j.anihpc.2013.09.003
LA  - en
ID  - AIHPC_2014__31_6_1289_0
ER  - 
%0 Journal Article
%A Adami, Riccardo
%A Cacciapuoti, Claudio
%A Finco, Domenico
%A Noja, Diego
%T Constrained energy minimization and orbital stability for the NLS equation on a star graph
%J Annales de l'I.H.P. Analyse non linéaire
%D 2014
%P 1289-1310
%V 31
%N 6
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.003/
%R 10.1016/j.anihpc.2013.09.003
%G en
%F AIHPC_2014__31_6_1289_0
Adami, Riccardo; Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego. Constrained energy minimization and orbital stability for the NLS equation on a star graph. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1289-1310. doi : 10.1016/j.anihpc.2013.09.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2013.09.003/

[1] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Fast solitons on star graphs, Rev. Math. Phys. 23 no. 4 (2011), 409 -451 | MR | Zbl

[2] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, On the structure of critical energy levels for the cubic focusing NLS on star graphs, J. Phys. A, Math. Theor. 45 (2012) | MR | Zbl

[3] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Stationary states of NLS on star graphs, Europhys. Lett. 100 (2012)

[4] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, arXiv:1206.5201 | MR | Zbl

[5] R. Adami, D. Noja, N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discrete Contin. Dyn. Syst., Ser. B 18 no. 5 (2013), 1155 -1188 | MR | Zbl

[6] V. Banica, L. Ignat, Dispersion for the Schrödinger equation on networks, J. Math. Phys. 52 (2011), 083703 | MR | Zbl

[7] G. Berkolaiko, R. Carlson, S. Fulling, P. Kuchment, Quantum Graphs and Their Applications, Contemp. Math. vol. 415 , American Mathematical Society, Providence, RI (2006) | MR

[8] J. Blank, P. Exner, M. Havlicek, Hilbert Spaces Operators in Quantum Physics, Springer, New York (2008) | MR

[9] J. Bona, R.C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q. 16 (2008), 1 -18 | MR | Zbl

[10] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983) | MR

[11] F. Camilli, C. Marchi, D. Schieborn, The vanishing viscosity limit for Hamilton–Jacobi equations on networks, arXiv:1207.6535 | MR | Zbl

[12] S. Cardanobile, D. Mugnolo, Analysis of FitzHugh–Nagumo–Rall model of a neuronal network, Math. Methods Appl. Sci. 30 (2007), 2281 -2308 | MR | Zbl

[13] R.C. Cascaval, C.T. Hunter, Linear and nonlinear Schrödinger equations on simple networks, Libertas Math. 30 (2010), 85 -98 | MR | Zbl

[14] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10 , American Mathematical Society, Providence, RI (2003) | MR | Zbl

[15] T. Cazenave, An Introduction to Semilinear Elliptic Equations, Editora do IM-UFRJ, Rio de Janeiro (2006)

[16] T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys. 85 (1982), 549 -561 | MR | Zbl

[17] P. Exner, Weakly coupled states on branching graphs, Lett. Math. Phys. 38 (1996), 313 -320 | MR | Zbl

[18] P. Exner, J.P. Keating, P. Kuchment, T. Sunada, A. Teplyaev, Analysis on Graphs and Its Applications, Proc. Symp. Pure Math. vol. 77 , American Mathematical Society, Providence, RI (2008) | MR

[19] L. Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier 55 no. 1 (2005), 199 -211 | EuDML | Numdam | MR | Zbl

[20] Z. Gang, I. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potentials, Adv. Math. 216 (2007), 443 -490 | MR | Zbl

[21] S. Gnutzman, U. Smilansky, S. Derevyanko, Stationary scattering from a nonlinear network, Phys. Rev. A 83 (2011), 033831

[22] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry I, J. Funct. Anal. 94 (1987), 308 -348 | MR | Zbl

[23] M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal. 74 (1987), 160 -197 | MR | Zbl

[24] S. Gustafson, K. Nakanishi, T. Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, Int. Math. Res. Not. 66 (2004), 3559 -3584 | MR | Zbl

[25] V. Kostrykin, R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, Math. Gen. 32 no. 4 (1999), 595 -630 | MR | Zbl

[26] P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media 14 no. 1 (2004), S107 -S128 | MR | Zbl

[27] P. Kuchment, Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A, Math. Gen. 38 no. 22 (2005), 4887 -4900 | MR | Zbl

[28] E.H. Lieb, M. Loss, Analysis, Grad. Stud. Math. vol. 14 , American Mathematical Society, Providence, RI (2001) | MR

[29] A.E. Miroshnichenko, M.I. Molina, Y.S. Kivshar, Localized modes and bistable scattering in nonlinear network junctions, Phys. Rev. Lett. 75 (2007), 04602

[30] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Math. Appl. vol. 53 , Kluwer Academic Publishers, Dordrecht, Boston, London (1991) | MR | Zbl

[31] Z. Sobirov, D. Matrasulov, K. Sabirov, S. Sawada, K. Nakamura, Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices, Phys. Rev. E 81 (2010), 066602 | MR

[32] K. Tintarev, K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London (2007) | MR | Zbl

[33] A. Tokuno, M. Oshikawa, E. Demler, Dynamics of the one dimensional Bose liquids: Andreev-like reflection at Y-junctions and the absence of Aharonov–Bohm effect, Phys. Rev. Lett. 100 (2008), 140402

[34] M. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39 (1986), 51 -68 | MR | Zbl

Cité par Sources :