Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 6 (2019), 1597-1612.

Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs

Riccardo Adami, Simone Dovetta, Enrico Serra, and Paolo Tilli

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We investigate the existence of ground states for the focusing nonlinear Schrödinger equation on a prototypical doubly periodic metric graph. When the nonlinearity power is below 4, ground states exist for every value of the mass, while, for every nonlinearity power between 4 (included) and 6 (excluded), a mark of L2-criticality arises, as ground states exist if and only if the mass exceeds a threshold value that depends on the power. This phenomenon can be interpreted as a continuous transition from a two-dimensional regime, for which the only critical power is 4, to a one-dimensional behavior, in which criticality corresponds to the power 6. We show that such a dimensional crossover is rooted in the coexistence of one-dimensional and two-dimensional Sobolev inequalities, leading to a new family of Gagliardo–Nirenberg inequalities that account for this continuum of critical exponents.

Article information

Anal. PDE, Volume 12, Number 6 (2019), 1597-1612.

Received: 7 May 2018
Revised: 7 September 2018
Accepted: 25 October 2018
First available in Project Euclid: 12 March 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35R02: Partial differential equations on graphs and networks (ramified or polygonal spaces)

metric graphs Sobolev inequalities threshold phenomena nonlinear Schrödinger equation


Adami, Riccardo; Dovetta, Simone; Serra, Enrico; Tilli, Paolo. Dimensional crossover with a continuum of critical exponents for NLS on doubly periodic metric graphs. Anal. PDE 12 (2019), no. 6, 1597--1612. doi:10.2140/apde.2019.12.1597.

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  • R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, “Fast solitons on star graphs”, Rev. Math. Phys. 23:4 (2011), 409–451.
  • R. Adami, E. Serra, and P. Tilli, “Lack of ground state for NLSE on bridge-type graphs”, pp. 1–11 in Mathematical technology of networks, edited by D. Mugnolo, Springer Proc. Math. Stat. 128, Springer, 2015.
  • R. Adami, E. Serra, and P. Tilli, “NLS ground states on graphs”, Calc. Var. Partial Differential Equations 54:1 (2015), 743–761.
  • R. Adami, E. Serra, and P. Tilli, “Threshold phenomena and existence results for NLS ground states on metric graphs”, J. Funct. Anal. 271:1 (2016), 201–223.
  • R. Adami, E. Serra, and P. Tilli, “Negative energy ground states for the $L^2$-critical NLSE on metric graphs”, Comm. Math. Phys. 352:1 (2017), 387–406.
  • R. Adami, E. Serra, and P. Tilli, “Nonlinear dynamics on branched structures and networks”, Riv. Math. Univ. Parma $($N.S.$)$ 8:1 (2017), 109–159.
  • F. Ali Mehmeti, Nonlinear waves in networks, Mathematical Research 80, Akademie, Berlin, 1994.
  • F. Ali Mehmeti, J. von Below, and S. Nicaise (editors), Partial differential equations on multistructures (Luminy, 1999), Lecture Notes in Pure and Applied Mathematics 219, Marcel Dekker, New York, 2001.
  • G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs 186, American Mathematical Society, Providence, RI, 2013.
  • H. Brezis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals”, Proc. Amer. Math. Soc. 88:3 (1983), 486–490.
  • C. Cacciapuoti, D. Finco, and D. Noja, “Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph”, Phys. Rev. E $(3)$ 91:1 (2015), art. id. 013206.
  • V. Caudrelier, “On the inverse scattering method for integrable PDEs on a star graph”, Comm. Math. Phys. 338:2 (2015), 893–917.
  • T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, New York, 2003.
  • P. Exner, “Contact interactions on graph superlattices”, J. Phys. A 29:1 (1996), 87–102.
  • P. Exner and R. Gawlista, “Band spectra of rectangular graph superlattices”, Phys. Rev. B 53:11 (1996), 7275–7286.
  • P. Exner and O. Turek, “High-energy asymptotics of the spectrum of a periodic square lattice quantum graph”, J. Phys. A 43:47 (2010), art. id. 474024.
  • S. Gnutzmann and D. Waltner, “Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory”, Phys. Rev. E 93:3 (2016), art. id. 032204.
  • G. Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics 105, American Mathematical Society, Providence, RI, 2009.
  • D. Noja, “Nonlinear Schrödinger equation on graphs: recent results and open problems”, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372:2007 (2014), art. id. 20130002.
  • D. Noja, D. Pelinovsky, and G. Shaikhova, “Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph”, Nonlinearity 28:7 (2015), 2343–2378.
  • A. Pankov, “Nonlinear Schrödinger equations on periodic metric graphs”, Discrete Contin. Dyn. Syst. 38:2 (2018), 697–714.
  • D. Pelinovsky and G. Schneider, “Bifurcations of standing localized waves on periodic graphs”, Ann. Henri Poincaré 18:4 (2017), 1185–1211.
  • O. Post, Spectral analysis on graph-like spaces, Lecture Notes in Mathematics 2039, Springer, 2012.
  • K. Ruedenberg and C. W. Scherr, “Free-electron network model for conjugated systems, I: Theory”, J. Chem. Phys. 21:9 (1953), 1565–1581.
  • K. K. Sabirov, Z. A. Sobirov, D. Babajanov, and D. U. Matrasulov, “Stationary nonlinear Schrödinger equation on simplest graphs”, Phys. Lett. A 377:12 (2013), 860–865.
  • Z. Sobirov, D. Matrasulov, K. Sabirov, S. Sawada, and K. Nakamura, “Integrable nonlinear Schrödinger equation on simple networks: connection formula at vertices”, Phys. Rev. E $(3)$ 81:6 (2010), art. id. 066602.
  • M. I. Weinstein, “Excitation thresholds for nonlinear localized modes on lattices”, Nonlinearity 12:3 (1999), 673–691.