Solitary waves for Maxwell-Dirac and Coulomb-Dirac models ,
, Ann. Inst. H. Poincaré Phys. Théor, vol.68, issue.2, pp.229-244, 1998.
Nonlinear waves in shallow honeycomb lattices, SIAM J. Appl. Math, vol.72, issue.1, pp.240-260, 2012. ,
, NLS ground states on graphs. Calc. Var. Partial Differential Equations, vol.54, pp.743-761, 2015.
Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal, vol.271, issue.1, pp.201-223, 2016. ,
Multiple positive bound states for the subcritical nls equation on metric graphs, 2017. ,
Negative energy ground states for the L 2-critical NLSE on metric graphs, Comm. Math. Phys, vol.352, issue.1, pp.387-406, 2017. ,
DOI : 10.1007/s00220-016-2797-2
URL : http://arxiv.org/pdf/1605.07666
Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol.140, 2003. ,
Dirac-Krein systems on star graphs, Integral Equations Operator Theory, vol.86, pp.121-150, 2016. ,
DOI : 10.1007/s00020-016-2311-4
URL : http://arxiv.org/pdf/1608.05865
Boundary conditions for dirac fermions on a terminated honeycomb lattice, Phys. Rev. B, vol.77, p.85423, 2008. ,
DOI : 10.1103/physrevb.77.085423
URL : http://arxiv.org/pdf/0710.2723
A remark on Krein's resolvent formula and boundary conditions, J. Phys. A, vol.38, issue.22, pp.4859-4864, 2005. ,
DOI : 10.1088/0305-4470/38/22/010
URL : http://arxiv.org/pdf/math-ph/0408021
Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys, vol.258, issue.1, pp.1-22, 2005. ,
A note on the problem ??u = ?u + u|u| 2 * ?2, Manuscripta Math, vol.54, issue.4, pp.373-379, 1986. ,
Nonlinear analysis and semilinear elliptic problems, Cambridge Studies in Advanced Mathematics, vol.104, 2007. ,
DOI : 10.1017/cbo9780511618260
Interpolation between Hilbert spaces, 2014. ,
A spinorial analogue of Aubin's inequality, Math. Z, vol.260, issue.1, pp.127-151, 2008. ,
DOI : 10.1007/s00209-007-0266-5
URL : http://arxiv.org/pdf/math/0308107
Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures, J. Math. Phys, vol.59, issue.1, p.11509, 2018. ,
Existence of excited states for a nonlinear Dirac field, Comm. Math. Phys, vol.119, issue.1, pp.153-176, 1988. ,
The cubic Dirac equation: small initial data in H 1 2 (R 2 ), Comm. Math. Phys, vol.343, issue.2, pp.515-562, 2016. ,
DOI : 10.1007/s00220-014-2164-0
URL : http://arxiv.org/pdf/1310.5280
Self-adjointness of two-dimensional Dirac operators on domains ,
DOI : 10.1007/s00023-017-0554-5
URL : http://arxiv.org/pdf/1704.06106
, , vol.18, pp.1371-1383, 2017.
Spectral gaps of Dirac operators describing graphene quantum dots ,
, Math. Phys. Anal. Geom, vol.20, issue.2, p.12, 2017.
Introduction to quantum graphs, Mathematical Surveys and Monographs, vol.186, 2013. ,
Neutrino billiards: time-reversal symmetrybreaking without magnetic fields, Proc. Roy. Soc. London Ser. A, vol.412, pp.53-74, 1842. ,
On the point spectrum of Dirac operators, J. Funct. Anal, vol.71, issue.2, pp.309-338, 1987. ,
Spectral statistics for the Dirac operator on graphs, J. Phys. A, vol.36, issue.11, pp.2747-2769, 2003. ,
The Calderón projection: new definition and applications, J. Geom. Phys, vol.59, issue.7, pp.784-826, 2009. ,
Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity, J. Differential Equations, vol.263, issue.11, pp.7941-7964, 2017. ,
DOI : 10.1016/j.jde.2017.08.029
URL : https://hal.archives-ouvertes.fr/hal-01540930
Multiple solutions for a self-consistent Dirac equation in two dimensions, J. Math. Phys, vol.59, issue.4, p.13, 2018. ,
DOI : 10.1063/1.5005998
URL : https://hal.archives-ouvertes.fr/hal-01590235
Weakly localized states for nonlinear Dirac equations, Calc. Var. Partial Differential Equations, vol.57, issue.6, 2018. ,
DOI : 10.1007/s00526-018-1420-0
URL : https://hal.archives-ouvertes.fr/hal-01709328
Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit, 2018. ,
URL : https://hal.archives-ouvertes.fr/hal-01842039
Global well-posedness for the massless cubic Dirac equation, Int. Math. Res. Not. IMRN, vol.22, issue.22, pp.6735-6828, 2016. ,
DOI : 10.1093/imrn/rnv361
URL : https://www.pure.ed.ac.uk/ws/files/30880508/Global_well_posedness_for_the_massless_cubic_Dirac_equation.pdf
On spectral stability of the nonlinear Dirac equation, J. Funct. Anal, vol.271, issue.6, pp.1462-1524, 2016. ,
Functional analysis, Sobolev spaces and partial differential equations, 2011. ,
DOI : 10.1007/978-0-387-70914-7
The free Dirac operator on compact and noncompact graphs, J. Math. Phys, vol.31, issue.5, pp.1157-1163, 1990. ,
DOI : 10.1063/1.529025
The cubic nonlinear dirac equation, JournéesJournéeséquations aux dérivées partielles, p.10, 2012. ,
DOI : 10.5802/jedp.84
URL : http://jedp.cedram.org/cedram-bin/article/JEDP_2012____A1_0.pdf
Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph, Phys. Rev. E, vol.91, issue.3, p.13206, 2015. ,
DOI : 10.1103/physreve.91.013206
Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, Nonlinearity, vol.30, issue.8, pp.3271-3303, 2017. ,
On the Majorana condition for nonlinear Dirac systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.35, issue.6, pp.1707-1717, 2018. ,
On the spectral theory of Gesztesy-? seba realizations of 1-D Dirac operators with point interactions on a discrete set, J. Differential Equations, vol.254, issue.9, pp.3835-3902, 2013. ,
Introduction to Dirac materials and topological insulators, Comptes Rendus Physique, vol.14, pp.760-778, 2013. ,
DOI : 10.1016/j.crhy.2013.09.012
URL : https://hal.archives-ouvertes.fr/hal-00942955
Existence of localized solutions for a classical nonlinear Dirac field, Comm. Math. Phys, vol.105, issue.1, pp.35-47, 1986. ,
DOI : 10.1007/bf01212340
Stability of solitary waves and vortices in a 2D nonlinear Dirac model, Phys. Rev. Lett, vol.116, issue.21, p.214101, 2016. ,
On the cubic Dirac equation with potential and the Lochak-Majorana condition, J. Math. Anal. Appl, vol.456, issue.2, pp.1203-1237, 2017. ,
Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal, vol.44, issue.6, pp.3755-3785, 2012. ,
Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys, vol.20, issue.8, pp.1007-1032, 2008. ,
Existence of infinitely many stationary solutions of the L 2subcritical and critical NLSE on compact metric graphs, J. Differential Equations, vol.264, issue.7, pp.4806-4821, 2018. ,
Ground states of the L 2-critical NLS equation with localized nonlinearity on a tadpole graph, 2018. ,
Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.19, issue.3 ,
, , 1990.
Analysis of models for quantum transport of electrons in graphene layers, Math. Models Methods Appl. Sci, vol.24, issue.11, pp.2287-2310, 2014. ,
URL : https://hal.archives-ouvertes.fr/hal-00850512
Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math, vol.167, issue.3, pp.515-614, 2007. ,
Nonrelativistic limit of the Dirac-Fock equations, Ann. Henri Poincaré, vol.2, issue.5, pp.941-961, 2001. ,
URL : https://hal.archives-ouvertes.fr/hal-00453389
Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations, Calc. Var. Partial Differential Equations, vol.4, issue.3, pp.265-281, 1996. ,
Variational methods in relativistic quantum mechanics, Bull. Amer. Math. Soc. (N.S.), vol.45, issue.4, pp.535-593, 2008. ,
URL : https://hal.archives-ouvertes.fr/hal-00156710
Symmetric ground states for a stationary relativistic mean-field model for nucleons in the non-relativistic limit, Rev. Math. Phys, vol.24, issue.10, p.30, 2012. ,
Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys, vol.171, issue.2, pp.323-350, 1995. ,
Partial differential equations, Graduate Studies in Mathematics, vol.19, 2010. ,
Honeycomb lattice potentials and dirac points, J. Amer. Math. Soc, vol.25, issue.4, pp.1169-1220, 2012. ,
Wave packets in honeycomb structures and two-dimensional Dirac equations, Comm. Math. Phys, vol.326, issue.1, pp.251-286, 2014. ,
Possible lattice distortions in the hubbard model for graphene, Phys. Rev. Lett, vol.107, p.66801, 2011. ,
Spectra of graphene nanoribbons with armchair and zigzag boundary conditions, Rev. Math. Phys, vol.26, issue.10, p.1450018, 2014. ,
Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol.25, 1997. ,
Dirac fermions in graphene and analogues: magnetic field and topological properties, 2013. ,
Locally finite extensions and Gesztesy-Seba realizations for the Dirac operator on a metric graph, 2018. ,
Elliptic partial differential equations of second order, Classics in Mathematics, 2001. ,
DOI : 10.1007/978-3-642-96379-7
Stationary scattering from a nonlinear network, Phys. Rev. A, vol.83, issue.3, p.33831, 2011. ,
DOI : 10.1103/physreva.83.033831
URL : http://orca.cf.ac.uk/17723/1/Stationary_scattering_from_a_nonlinear_network.pdf
Stationary waves on nonlinear quantum graphs: general framework and canonical perturbation theory, Phys. Rev. E, vol.93, issue.3, p.32204, 2016. ,
DOI : 10.1103/physreve.93.032204
URL : http://arxiv.org/pdf/1510.00351
Application of the multiparameter theory of perturbations of Fredholm operators to Bloch functions, Mat. Zametki, vol.86, issue.6, pp.819-828, 2009. ,
Ground state properties of graphene in Hartree-Fock theory, J. Math. Phys, vol.53, issue.9, p.27, 2012. ,
URL : https://hal.archives-ouvertes.fr/hal-00716470
Model for a quantum hall effect without landau levels: Condensed-matter realization of the "parity anomaly, Phys. Rev. Lett, vol.61, pp.2015-2018, 1988. ,
Possible realization of directional optical waveguides in photonic lattice crystals with broken time-reversal symmetry, Phys. Rev. Lett, vol.100, p.13904, 2008. ,
Band-edge solitons, nonlinear Schrödinger/GrossPitaevskii equations, and effective media, Multiscale Model. Simul, vol.8, issue.4, pp.1055-1101, 2010. ,
DOI : 10.1137/090769417
URL : http://arxiv.org/pdf/1002.1986
Existence results for solutions to nonlinear Dirac equations on compact spin manifolds, Manuscripta Math, vol.135, issue.3-4, pp.329-360, 2011. ,
DOI : 10.1007/s00229-010-0417-6
Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds, J. Funct. Anal, vol.260, issue.1, pp.253-307, 2011. ,
Riemannian geometry and geometric analysis, 2011. ,
DOI : 10.1007/978-3-642-21298-7
URL : http://cds.cern.ch/record/1666885/files/9783540773405_TOC.pdf
Quantum spin hall effect in graphene, Phys. Rev. Lett, vol.95, p.226801, 2005. ,
Graphene: New bridge between condensed matter physics and quantum electrodynamics, Solid State Communications, vol.143, issue.1, pp.3-13, 2007. ,
DOI : 10.1016/j.ssc.2007.02.043
URL : http://arxiv.org/pdf/cond-mat/0703374
Introduction to Solid State Physics, 2004. ,
Kirchhoff's rule for quantum wires, J. Phys. A, vol.32, issue.4, pp.595-630, 1999. ,
DOI : 10.1088/0305-4470/32/4/006
URL : http://arxiv.org/pdf/math-ph/9806013
Quantum graphs. I. Some basic structures, Waves Random Media, vol.14, issue.1, pp.107-128, 2004. ,
On the spectra of carbon nano-structures, Comm. Math. Phys, vol.275, issue.3, pp.805-826, 2007. ,
Waves in honeycomb structures, JournéesJournéeséquations aux dérivées partielles, p.10, 2012. ,
Symmetric excited states for a mean-field model for a nucleon, J. Differential Equations, vol.255, issue.10, pp.3536-3563, 2013. ,
Ground states of nonlinear Schrödinger equation on star metric graphs, J. Math. Anal. Appl, vol.459, issue.2, pp.661-685, 2018. ,
, Graduate Studies in Mathematics, vol.14, 2001.
The concentration-compactness principle in the calculus of variations. The locally compact case, I. Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.1, issue.2, pp.109-145, 1984. ,
The concentration-compactness principle in the calculus of variations. The limit case, I. Rev. Mat. Iberoamericana, vol.1, issue.1, pp.145-201, 1985. ,
The concentration-compactness principle in the calculus of variations. The limit case, II. Rev. Mat. Iberoamericana, vol.1, issue.2, pp.45-121, 1985. ,
Generalized effective hamiltonian for graphene under nonuniform strain, Phys. Rev. B, vol.88, p.155405, 2013. ,
Ground state on the dumbbell graph, Appl. Math. Res. Express. AMRX, vol.1, pp.98-145, 2016. ,
Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol.74, 1989. ,
Symmetry of boundary conditions of the dirac equation for electrons in carbon nanotubes, Journal of Physics: Condensed Matter, vol.16, issue.13, p.2371, 2004. ,
Radial solutions of ?u + f (u) = 0 with prescribed numbers of zeros, J. Differential Equations, vol.83, issue.2, pp.368-378, 1990. ,
Translated from the French by J. Potter, vol.II, 1962. ,
Nonlinear optics, 2004. ,
Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, vol.372, 2007. ,
Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, vol.372, 2007. ,
Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity, vol.28, issue.7, pp.2343-2378, 2015. ,
Standing waves for the nls on the double-bridge graph and a rational-irrational dichotomy, 2017. ,
,
Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett, vol.98, p.103901, 2007. ,
Bose-Einstein condensation, International Series of Monographs on Physics, vol.116, 2003. ,
Self-adjoint extensions of restrictions, Oper. Matrices, vol.2, issue.4, pp.483-506, 2008. ,
Equilateral quantum graphs and boundary triples, Analysis on graphs and its applications, vol.77, pp.469-490, 2008. ,
DOI : 10.1090/pspum/077/2459887
URL : http://www.math.hu-berlin.de/~post/papers/cambridge-proc.pdf
Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, vol.65, 1986. ,
Methods of modern mathematical physics. I. Functional analysis, 1972. ,
Methods of modern mathematical physics. IV. Analysis of operators, 1978. ,
Introduction to various aspects of degree theory in Banach spaces, Mathematical Surveys and Monographs, 1986. ,
,
, Dynamics of dirac solitons in networks, 2017.
Modern quantum mechanics; rev, 1994. ,
A remark on boundary value problems for the Dirac operator, Quart. J. Math. Oxford Ser, vol.46, issue.2, pp.509-516, 1995. ,
, MSE 532 Computational Material Science
Condensed-matter simulation of a three-dimensional anomaly, Phys. Rev. Lett, vol.53, pp.2449-2452, 1984. ,
Bound states of the NLS equation on metric graphs with localized nonlinearities, J. Differential Equations, vol.260, issue.7, pp.5627-5644, 2016. ,
On the lack of bound states for certain NLS equations on metric graphs, Nonlinear Anal, vol.145, pp.68-82, 2016. ,
Infinite mass boundary conditions for dirac operators, Journal of Spectral Theory, 2016. ,
Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol.34 ,
Applications to nonlinear partial differential equations and Hamiltonian systems, 2008. ,
The method of Nehari manifold, Handbook of nonconvex analysis and applications, pp.597-632, 2010. ,
Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, vol.106, 2006. ,
NLS ground states on metric graphs with localized nonlinearities, J. Math. Anal. Appl, vol.433, issue.1, pp.291-304, 2016. ,
The Dirac equation. Texts and Monographs in Physics, 1992. ,
Linear and nonlinear photonic jackiw-rebbi states in interfaced binary waveguide arrays, Phys. Rev. A, vol.96, p.13831, 2017. ,
Optical analogue of relativistic dirac solitons in binary waveguide arrays, Annals of Physics, vol.340, issue.1, pp.179-187, 2014. ,
Dirac solitons in square binary waveguide lattices, Phys. Rev. A, vol.91, p.23814, 2015. ,
Spectrum of graphene quantum dots, 2017. ,
The band theory of graphite, Phys. Rev, vol.71, pp.622-634, 1947. ,
A remark on nonlinear dirac equation, Proceed.of the AMS, vol.138, issue.10, pp.3753-3758, 2010. ,
Minimax theorems, Nonlinear Differential Equations and their Applications, vol.24, 1996. ,