S. Abenda, Solitary waves for Maxwell-Dirac and Coulomb-Dirac models

, Ann. Inst. H. Poincaré Phys. Théor, vol.68, issue.2, pp.229-244, 1998.

J. Mark, Y. Ablowitz, and . Zhu, Nonlinear waves in shallow honeycomb lattices, SIAM J. Appl. Math, vol.72, issue.1, pp.240-260, 2012.

R. Adami, E. Serra, and P. Tilli, NLS ground states on graphs. Calc. Var. Partial Differential Equations, vol.54, pp.743-761, 2015.

R. Adami, E. Serra, and P. Tilli, Threshold phenomena and existence results for NLS ground states on metric graphs, J. Funct. Anal, vol.271, issue.1, pp.201-223, 2016.

R. Adami, E. Serra, and P. Tilli, Multiple positive bound states for the subcritical nls equation on metric graphs, 2017.

R. Adami, E. Serra, and P. Tilli, 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

A. Robert, J. J. Adams, and . Fournier, Sobolev spaces, Pure and Applied Mathematics (Amsterdam), vol.140, 2003.

V. Adamyan, H. Langer, C. Tretter, and M. Winklmeier, 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

A. R. Akhmerov and C. W. Beenakker, 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

S. Albeverio and K. Pankrashkin, 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

G. Allaire and A. Piatnitski, Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys, vol.258, issue.1, pp.1-22, 2005.

A. Ambrosetti and M. Struwe, A note on the problem ??u = ?u + u|u| 2 * ?2, Manuscripta Math, vol.54, issue.4, pp.373-379, 1986.

A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge Studies in Advanced Mathematics, vol.104, 2007.
DOI : 10.1017/cbo9780511618260

Y. Ameur, Interpolation between Hilbert spaces, 2014.

B. Ammann, J. Grosjean, E. Humbert, and B. Morel, 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

J. Arbunich and C. Sparber, Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures, J. Math. Phys, vol.59, issue.1, p.11509, 2018.

M. Balabane, T. Cazenave, A. Douady, and F. Merle, Existence of excited states for a nonlinear Dirac field, Comm. Math. Phys, vol.119, issue.1, pp.153-176, 1988.

I. Bejenaru and S. Herr, 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

R. D. Benguria, S. Fournais, E. Stockmeyer, H. Van-den, and . Bosch, Self-adjointness of two-dimensional Dirac operators on domains
DOI : 10.1007/s00023-017-0554-5

URL : http://arxiv.org/pdf/1704.06106

H. Poincaré, , vol.18, pp.1371-1383, 2017.

R. D. Benguria, S. Fournais, E. Stockmeyer, H. Van-den, and . Bosch, Spectral gaps of Dirac operators describing graphene quantum dots

, Math. Phys. Anal. Geom, vol.20, issue.2, p.12, 2017.

G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol.186, 2013.

M. V. Berry and R. J. Mondragon, Neutrino billiards: time-reversal symmetrybreaking without magnetic fields, Proc. Roy. Soc. London Ser. A, vol.412, pp.53-74, 1842.

A. Berthier and V. Georgescu, On the point spectrum of Dirac operators, J. Funct. Anal, vol.71, issue.2, pp.309-338, 1987.

J. Bolte and J. Harrison, Spectral statistics for the Dirac operator on graphs, J. Phys. A, vol.36, issue.11, pp.2747-2769, 2003.

M. Bernhelm-booß-bavnbek, C. Lesch, and . Zhu, The Calderón projection: new definition and applications, J. Geom. Phys, vol.59, issue.7, pp.784-826, 2009.

W. Borrelli, 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

W. Borrelli, 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

W. Borrelli, 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

W. Borrelli, R. Carlone, and L. Tentarelli, Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01842039

N. Bournaveas and T. Candy, 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

N. Boussa¨?dboussa¨?d and A. Comech, On spectral stability of the nonlinear Dirac equation, J. Funct. Anal, vol.271, issue.6, pp.1462-1524, 2016.

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, 2011.
DOI : 10.1007/978-0-387-70914-7

W. Bulla and T. Trenkler, 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

F. Cacciafesta, 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

C. Cacciapuoti, D. Finco, and D. Noja, 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

C. Cacciapuoti, D. Finco, and D. Noja, 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.

T. Candy and S. Herr, On the Majorana condition for nonlinear Dirac systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.35, issue.6, pp.1707-1717, 2018.

R. Carlone, M. Malamud, and A. Posilicano, 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.

J. Cayssol, 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

T. Cazenave and L. Vázquez, 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

J. Cuevas-maraver, P. G. Kevrekidis, A. Saxena, A. Comech, and R. Lan, Stability of solitary waves and vortices in a 2D nonlinear Dirac model, Phys. Rev. Lett, vol.116, issue.21, p.214101, 2016.

M. Piero-d'ancona and . Okamoto, On the cubic Dirac equation with potential and the Lochak-Majorana condition, J. Math. Anal. Appl, vol.456, issue.2, pp.1203-1237, 2017.

Y. Ding and B. Ruf, Existence and concentration of semiclassical solutions for Dirac equations with critical nonlinearities, SIAM J. Math. Anal, vol.44, issue.6, pp.3755-3785, 2012.

Y. Ding and J. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys, vol.20, issue.8, pp.1007-1032, 2008.

S. Dovetta, 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.

S. Dovetta and L. Tentarelli, Ground states of the L 2-critical NLS equation with localized nonlinearity on a tadpole graph, 2018.

I. Ekeland, Convexity methods in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol.19, issue.3

. Springer-verlag, , 1990.

R. E. Hajj and F. Méhats, 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

B. László-erd?-os, H. Schlein, and . Yau, Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math, vol.167, issue.3, pp.515-614, 2007.

M. J. Esteban and E. Séré, 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

M. J. Esteban, V. Georgiev, and E. Séré, 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.

M. J. Esteban, M. Lewin, and E. Séré, 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

M. J. Esteban and S. R. Nodari, 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.

M. J. Esteban and . Séré, Stationary states of the nonlinear Dirac equation: a variational approach, Comm. Math. Phys, vol.171, issue.2, pp.323-350, 1995.

C. Lawrence and . Evans, Partial differential equations, Graduate Studies in Mathematics, vol.19, 2010.

L. Charles, M. I. Fefferman, and . Weinstein, Honeycomb lattice potentials and dirac points, J. Amer. Math. Soc, vol.25, issue.4, pp.1169-1220, 2012.

L. Charles, M. I. Fefferman, and . Weinstein, Wave packets in honeycomb structures and two-dimensional Dirac equations, Comm. Math. Phys, vol.326, issue.1, pp.251-286, 2014.

R. L. Frank and E. H. Lieb, Possible lattice distortions in the hubbard model for graphene, Phys. Rev. Lett, vol.107, p.66801, 2011.

P. Freitas and P. Siegl, Spectra of graphene nanoribbons with armchair and zigzag boundary conditions, Rev. Math. Phys, vol.26, issue.10, p.1450018, 2014.

T. Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol.25, 1997.

J. Fuchs, Dirac fermions in graphene and analogues: magnetic field and topological properties, 2013.

H. Gernandt and C. Trunk, Locally finite extensions and Gesztesy-Seba realizations for the Dirac operator on a metric graph, 2018.

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, 2001.
DOI : 10.1007/978-3-642-96379-7

S. Gnutzmann, U. Smilansky, and S. A. Derevyanko, 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

S. Gnutzmann and D. Waltner, 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

V. V. Grushin, Application of the multiparameter theory of perturbations of Fredholm operators to Bloch functions, Mat. Zametki, vol.86, issue.6, pp.819-828, 2009.

C. Hainzl, C. Mathieu-lewin, and . Sparber, 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

F. D. Haldane, 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.

F. D. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic lattice crystals with broken time-reversal symmetry, Phys. Rev. Lett, vol.100, p.13904, 2008.

B. Ilan and M. I. Weinstein, 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

T. Isobe, 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

T. Isobe, Nonlinear Dirac equations with critical nonlinearities on compact Spin manifolds, J. Funct. Anal, vol.260, issue.1, pp.253-307, 2011.

J. Jost, 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

C. L. Kane and E. J. Mele, Quantum spin hall effect in graphene, Phys. Rev. Lett, vol.95, p.226801, 2005.

M. I. Katsnelson and K. S. Novoselov, 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

C. Kittel, Introduction to Solid State Physics, 2004.

V. Kostrykin and R. Schrader, 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

P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media, vol.14, issue.1, pp.107-128, 2004.

P. Kuchment and O. Post, On the spectra of carbon nano-structures, Comm. Math. Phys, vol.275, issue.3, pp.805-826, 2007.

F. Charles, L. , and M. I. Weinstein, Waves in honeycomb structures, JournéesJournéeséquations aux dérivées partielles, p.10, 2012.

L. Lo¨?lo¨?, S. R. Treust, and . Nodari, Symmetric excited states for a mean-field model for a nucleon, J. Differential Equations, vol.255, issue.10, pp.3536-3563, 2013.

Y. Li, F. Li, and J. Shi, Ground states of nonlinear Schrödinger equation on star metric graphs, J. Math. Anal. Appl, vol.459, issue.2, pp.661-685, 2018.

H. Elliott, M. Lieb, and . Loss, Graduate Studies in Mathematics, vol.14, 2001.

P. Lions, 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.

P. Lions, The concentration-compactness principle in the calculus of variations. The limit case, I. Rev. Mat. Iberoamericana, vol.1, issue.1, pp.145-201, 1985.

P. Lions, The concentration-compactness principle in the calculus of variations. The limit case, II. Rev. Mat. Iberoamericana, vol.1, issue.2, pp.45-121, 1985.

J. L. Mañes, M. Fernando-de-juan, . Sturla, A. H. María, and . Vozmediano, Generalized effective hamiltonian for graphene under nonuniform strain, Phys. Rev. B, vol.88, p.155405, 2013.

J. L. Marzuola and D. E. Pelinovsky, Ground state on the dumbbell graph, Appl. Math. Res. Express. AMRX, vol.1, pp.98-145, 2016.

J. Mawhin and M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol.74, 1989.

E. Mccann and V. Ko, 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.

K. Mcleod, W. C. Troy, and F. B. Weissler, Radial solutions of ?u + f (u) = 0 with prescribed numbers of zeros, J. Differential Equations, vol.83, issue.2, pp.368-378, 1990.

A. Messiah, Translated from the French by J. Potter, vol.II, 1962.

J. Moloney and A. Newell, Nonlinear optics, 2004.

D. Noja, Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, vol.372, 2007.

D. Noja, Nonlinear Schrödinger equation on graphs: recent results and open problems, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, vol.372, 2007.

D. Noja, D. Pelinovsky, and G. Shaikhova, Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph, Nonlinearity, vol.28, issue.7, pp.2343-2378, 2015.

D. Noja, S. Rolando, and S. Secchi, Standing waves for the nls on the double-bridge graph and a rational-irrational dichotomy, 2017.

O. Peleg, G. Bartal, B. Freedman, O. Manela, and M. Segev,

N. Demetrios and . Christodoulides, Conical diffraction and gap solitons in honeycomb photonic lattices, Phys. Rev. Lett, vol.98, p.103901, 2007.

L. Pitaevskii and S. Stringari, Bose-Einstein condensation, International Series of Monographs on Physics, vol.116, 2003.

A. Posilicano, Self-adjoint extensions of restrictions, Oper. Matrices, vol.2, issue.4, pp.483-506, 2008.

O. Post, 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

H. Paul and . Rabinowitz, 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.

M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, 1972.

M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, 1978.

E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, Mathematical Surveys and Monographs, 1986.

K. Karimjon, D. Sabirov, D. U. Babajanov, . Matrasulov, and G. Panagiotis,

. Kevrekidis, Dynamics of dirac solitons in networks, 2017.

J. Sakurai, Modern quantum mechanics; rev, 1994.

K. M. Schmidt, 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

G. W. Semenoff, Condensed-matter simulation of a three-dimensional anomaly, Phys. Rev. Lett, vol.53, pp.2449-2452, 1984.

E. Serra and L. Tentarelli, Bound states of the NLS equation on metric graphs with localized nonlinearities, J. Differential Equations, vol.260, issue.7, pp.5627-5644, 2016.

E. Serra and L. Tentarelli, On the lack of bound states for certain NLS equations on metric graphs, Nonlinear Anal, vol.145, pp.68-82, 2016.

E. Stockmeyer and S. Vugalter, Infinite mass boundary conditions for dirac operators, Journal of Spectral Theory, 2016.

M. Struwe, Variational methods, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol.34

. Springer-verlag, Applications to nonlinear partial differential equations and Hamiltonian systems, 2008.

A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, pp.597-632, 2010.

T. Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, vol.106, 2006.

L. Tentarelli, NLS ground states on metric graphs with localized nonlinearities, J. Math. Anal. Appl, vol.433, issue.1, pp.291-304, 2016.

B. Thaller, The Dirac equation. Texts and Monographs in Physics, 1992.

X. Truong, F. Tran, and . Biancalana, Linear and nonlinear photonic jackiw-rebbi states in interfaced binary waveguide arrays, Phys. Rev. A, vol.96, p.13831, 2017.

X. Truong, S. Tran, F. Longhi, and . Biancalana, Optical analogue of relativistic dirac solitons in binary waveguide arrays, Annals of Physics, vol.340, issue.1, pp.179-187, 2014.

X. Truong, X. N. Tran, F. Nguyen, and . Biancalana, Dirac solitons in square binary waveguide lattices, Phys. Rev. A, vol.91, p.23814, 2015.

H. Van-den and . Bosch, Spectrum of graphene quantum dots, 2017.

P. R. Wallace, The band theory of graphite, Phys. Rev, vol.71, pp.622-634, 1947.

C. Wang, A remark on nonlinear dirac equation, Proceed.of the AMS, vol.138, issue.10, pp.3753-3758, 2010.

M. Willem, Minimax theorems, Nonlinear Differential Equations and their Applications, vol.24, 1996.