Constrained energy minimization and orbital stability for the NLS equation on a star graph
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 6, p. 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 $i{\partial }_{t}\Psi \left(t\right)=-\Delta \Psi \left(t\right)-{|\Psi \left(t\right)|}^{2\mu }\Psi \left(t\right)+\alpha {\delta }_{0}\Psi \left(t\right)$, 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 $0<\mu ⩽2$ 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 ${\stackrel{ˆ}{\Psi }}_{m}\in {H}^{1}\left(𝒢\right)$.Moreover, the set of minimizers has the structure $ℳ=\left\{{e}^{i\theta }{\stackrel{ˆ}{\Psi }}_{m},\phantom{\rule{0.166667em}{0ex}}\theta \in ℝ\right\}$. Correspondingly, for every $m<{m}^{⁎}$ there exists a unique $\omega =\omega \left(m\right)$ such that the standing wave ${\stackrel{ˆ}{\Psi }}_{\omega }{e}^{i\omega t}$ 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 $\alpha =0$.

@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},
publisher = {Elsevier},
volume = {31},
number = {6},
year = {2014},
pages = {1289-1310},
doi = {10.1016/j.anihpc.2013.09.003},
zbl = {1304.81087},
mrnumber = {3280068},
language = {en},
url = {http://www.numdam.org/item/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, Volume 31 (2014) no. 6, pp. 1289-1310. doi : 10.1016/j.anihpc.2013.09.003. http://www.numdam.org/item/AIHPC_2014__31_6_1289_0/

[1] R. Adami, C. Cacciapuoti, D. Finco, D. Noja, Fast solitons on star graphs, Rev. Math. Phys. 23 no. 4 (2011), 409 -451 | MR 2804557 | Zbl 1222.35182

[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 2924493 | Zbl 1247.81104

[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 3260240 | Zbl 1300.35129

[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 3038749 | Zbl 1280.35132

[6] V. Banica, L. Ignat, Dispersion for the Schrödinger equation on networks, J. Math. Phys. 52 (2011), 083703 | MR 2858075 | Zbl 1272.81079

[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 2279143

[8] J. Blank, P. Exner, M. Havlicek, Hilbert Spaces Operators in Quantum Physics, Springer, New York (2008) | MR 2458485

[9] J. Bona, R.C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q. 16 (2008), 1 -18 | MR 2500096 | Zbl 1167.76054

[10] H. Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris (1983) | MR 697382

[11] F. Camilli, C. Marchi, D. Schieborn, The vanishing viscosity limit for Hamilton–Jacobi equations on networks, arXiv:1207.6535 | MR 3032299 | Zbl 1278.35256

[12] S. Cardanobile, D. Mugnolo, Analysis of FitzHugh–Nagumo–Rall model of a neuronal network, Math. Methods Appl. Sci. 30 (2007), 2281 -2308 | MR 2362954 | Zbl 1195.92007

[13] R.C. Cascaval, C.T. Hunter, Linear and nonlinear Schrödinger equations on simple networks, Libertas Math. 30 (2010), 85 -98 | MR 2675854 | Zbl 1210.35271

[14] T. Cazenave, Semilinear Schrödinger Equations, Courant Lect. Notes Math. vol. 10 , American Mathematical Society, Providence, RI (2003) | MR 2002047 | Zbl 1055.35003

[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 677997 | Zbl 0513.35007

[17] P. Exner, Weakly coupled states on branching graphs, Lett. Math. Phys. 38 (1996), 313 -320 | MR 1416707 | Zbl 0861.35068

[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 2459860

[19] L. Friedlander, Extremal properties of eigenvalues for a metric graph, Ann. Inst. Fourier 55 no. 1 (2005), 199 -211 | Numdam | MR 2141695 | Zbl 1074.34078

[20] Z. Gang, I. Sigal, Relaxation of solitons in nonlinear Schrödinger equations with potentials, Adv. Math. 216 (2007), 443 -490 | MR 2351368 | Zbl 1126.35065

[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 1081647 | Zbl 0711.58013

[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 901236 | Zbl 0656.35122

[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 2101699 | Zbl 1072.35167

[25] V. Kostrykin, R. Schrader, Kirchhoff's rule for quantum wires, J. Phys. A, Math. Gen. 32 no. 4 (1999), 595 -630 | MR 1671833 | Zbl 0928.34066

[26] P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media 14 no. 1 (2004), S107 -S128 | MR 2042548 | Zbl 1063.81058

[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 2148631 | Zbl 1070.81062

[28] E.H. Lieb, M. Loss, Analysis, Grad. Stud. Math. vol. 14 , American Mathematical Society, Providence, RI (2001) | MR 1817225

[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 1190927 | Zbl 0744.26011

[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 2736292

[32] K. Tintarev, K.-H. Fieseler, Concentration Compactness. Functional-Analytic Grounds and Applications, Imperial College Press, London (2007) | MR 2294665 | Zbl 1118.49001

[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 820338 | Zbl 0594.35005