Stefan Le Coz

Research area

Nonlinear Schrödinger and Klein-Gordon equations, standing waves, solitons, multi-solitons, orbital stability and instability, instability by blow-up, variational methods

Mentoring

Post-docs/visiting researchers

• 11/2023-11/2025 Boris Shakarov, CIMI post-doc
• 06/2018-05/2019 Zhong Wang (Foshan University, China), China Scholarship Council visitor

PhD students

• 09/2023-2026 Elio Durand-Simmonet (UPS Toulouse). co-advised with Romain Duboscq.
• 10/2019-06/2023 Perla Kfoury (UPS Toulouse). co-advised with Tai-Peng Tsai
• 10/2019-05/2022 Phan Van Tin (UPS Toulouse).
• 09/2017-11/2019 Elek Csobo (TU Delft, Netherlands). co-advised with François Genoud and Jan van Neerven

Publications

Preprints

• Ground states on a fractured strip and one dimensional reduction S. Le Coz and B. Shakarov, hal-04806174
• Ground States of the Nonlinear Schrödinger Equation on the Tadpole Graph with a Repulsive Delta Vertex Condition R. Duboscq, É. Durand-Simonnet, S. Le Coz, hal-05053744.
• Traveling waves in periodic metric graphs via spatial dynamics S. Le Coz, D. E. Pelinovsky, G. Schneider, hal-05054394.
• A minimal mass blow-up solution on a nonlinear quantum star graph, F. Genoud, S. Le Coz, and J. Royer, hal-04045570.

Software libary

• Grafidi, a Python package for the solution of Partial Differential Equations on complex networks, C. Besse, R. Duboscq, and S. Le Coz.

In journals

• Computation of excited states for the nonlinear Schrödinger equation: numerical and theoretical analysis C. Besse, R. Duboscq, S. Le Coz, ESAIM, Math. Model. Numer. Anal. 59(2):899--923, 2025.
• Variational properties of space-periodic standing waves of nonlinear Schrödinger equations with general nonlinearities P. Kfoury and S. Le Coz, ESAIM, Control Optim. Calc. Var. 30(79):31 pp. 2024.
• Analysis of stability and instability for standing waves of the double power one dimensional nonlinear Schrödinger equation, P. Kfoury, S. Le Coz, and T.-P. Tsai, Comptes Rendus Mathématique, 360:867--892, 2022. See also the Github repository.
• Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library, C. Besse, R. Duboscq, S. Le Coz, SMAI Journal of Computational Mathematics, 8:1--47, 2022.
• Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs, C. Besse, R. Duboscq, S. Le Coz, Annales Henri Lebesgue, 5:387--428, 2022.
• Stability of the multi-solitons of the modified Korteweg-de Vries equation, S. Le Coz and Z. Wang, Nonlinearity, 34(10):7109--7143, 2021.
• Stability of multi-solitons for the derivative nonlinear Schrödinger equation, S. Le Coz, Y. Wu, Int. Math. Res. Notices (2018), no. 13, 4120–4170
  
• Stability of Periodic Waves of 1D Cubic Nonlinear Schrödinger Equations , S. Gustafson, S. Le Coz, T.-P. Tsai, Appl. Math. Res. Express (2017), no. 2, 431–487
  
• On the Cauchy problem and the black solitons of a singularly perturbed Gross-Pitaevskii equation, I. Ianni, S. Le Coz, J. Royer, SIAM J. Math. Anal. 49 (2017), no. 2, 1060–1099
  
• Multi-speed solitary waves of nonlinear Schrödinger systems: theoretical and numerical analysis, F. Delebecque, S. Le Coz, R.-M. Weishäupl, Commun. Math. Sci. 14 (2016), no. 6, 1599–1624.
  
• Minimal mass blow up solutions for a double power nonlinear Schrödinger equation, S. Le Coz, Y. Martel, and P. Raphael, Rev. Mat. Iberoam. 32 (2016), no. 3, 795–833
  
• Fast-moving finite and infinite trains of solitons for nonlinear Schrödinger equations, S. Le Coz, D. Li, T.-P. Tsai, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1251-1282.
  
• Infinite soliton and kink-soliton trains for nonlinear Schrödinger equations, S. Le Coz, T.-P. Tsai, Nonlinearity, 27 (2014), 2689-2709.
  
• Multi-solitary waves for the nonlinear Klein-Gordon equation, J. Bellazzini, M. Ghimenti, and S. Le Coz, Comm. Partial Differential Equations, 39 (2014), no. 8, 1479–1522.
  
• Multi-Speed solitary wave solutions for nonlinear Schrödinger systems, I. Ianni and S. Le Coz, J. London Math. Soc. (2), 89 (2014), 623-639.
  
• On the stability of standing waves of Klein-Gordon equations in a semiclassical regime, M. Ghimenti, S. Le Coz, and M. Squassina, Discrete Contin. Dyn. Syst. - Series A, 33, no 6 (2013), 2389-2401.
  
• High speed excited multi-solitons in nonlinear Schrödinger equations, R. Côte and S. Le Coz, J. Math. Pures Appl. 96 (2011) 135-166.
  
• Orbital stability of standing waves of a semi-classical nonlinear Schrödinger-Poisson equation, I. Ianni and S. Le Coz, Advances in Differential Equations, 14, (2009), 717-748.
  
• Instability for standing waves of nonlinear Klein-Gordon equations via mountain-pass arguments, L. Jeanjean and S. Le Coz, Transactions of the AMS, 361, 10, (2009), 5401-5416.
  
• A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, S. Le Coz, Advanced Nonlinear Studies, 8, (2008), 455-463.
  
• Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Physica D, 237, (2008), 1103-1128.
  
• An existence and stability result for standing waves of nonlinear Schrödinger equations, L. Jeanjean and S. Le Coz, Advances in Differential Equations, 11, (2006), 813-840.

Book Chapter

• Standing waves in nonlinear Schrödinger equations, S. Le Coz, Analytical and Numerical Aspects of Partial Differential Equations, de Gruyter, Berlin, (2009), 151-192.

Proceedings

• Finite and infinite soliton and kink-soliton trains of nonlinear Schrödinger equations, S. Le Coz, T.-P. Tsai, Proceedings of the 6th International Congress of Chinese Mathematicians, Advanced Lectures in Mathematics, Volume 36, International Press, (2017), pp. 43–56.

Dissertations

• Existence et stabilité de solitons, multi-solitons et solutions explosives dans quelques équations dispersives non linéaires. S. Le Coz, Accreditation to Supervise Research (HDR), Université Toulouse 3, 2018.
• Existence, stability and instability of standing waves for nonlinear Schrödinger and Klein-Gordon equations. S. Le Coz, PhD Thesis, University of Franche-Comté, 2007.

Grants

• 2025-2026 Principal Investigator of the CNRS IEA Project Nonlinear Quantum Graphs (in collaboration with Dmitry Pelinovsky at McMaster, Hamilton, Canada)
• 2024-2028 Local Principal Inverstigator of the ANR Project NQG (National PI Louis Jeanjean)
• 2021-2023 Principal Investigator of the Labex CIMI Project COCASSE
• 2020-2021 Principal Investigator of the CNRS IEA Project Periodic Waves in Nonlinear Dispersive PDE (in collaboration with Tai-Peng Tsai at UBC, Vancouver, Canada)
• 2020-2022 Member of the CIMI Project PDEs on Quantum Graphs (PI Julien Royer)
• 2017-2020 Principal Investigator of the Labex CIMI Project Nonlinear Waves
• 2017-2018 Principal Investigator (french side) of the Dutch-French van Gogh Project NoWaves (François Genoud PI for the Dutch side)
• 2014-2018 Member of the ANR projet MAToS - Mathematical Analysis of Topological Singularities in some physical problems (PI Radu Ignat)
• 2014-2015 Member of the Toulouse Idex Emergence project NewTiTAnS - New Trends in The Analysis of topological Singularities in mathematical physics (PI Radu Ignat)
• 2014-2015 Member of the PHC Amadeus Project "Quantum transport models for semiconductors and Bose-Einstein condensates" (PI Claudia Negulescu)
• 2009-2013 Principal Investigator of the ANR project ESONSE - Excited States Of Nonlinear Schrödinger Equations

Workshop organisation

• Summer School on Nonlinear quantum graphs, Juin 2024, Valenciennes, France (with Colette de Coster, Damien Galand et Louis Jeanjean)
• Nonlinear Quantum Graphs Day, Janvier 2024, Toulouse, France (with Romain Duboscq)
• Numerical and theoretical advances in quantum mechanics, Janvier 2023, Toulouse, France (with Claudia Negulescu and Julien Royer)
• Nonlinear Waveguides and Related Topics, October 2018, Toulouse, France (with Julien Royer)
• Analysis of singular patterns in variational models, June 2018, Toulouse, France (with Radu Ignat and Mihai Maris)
• Second Meeting of the ANR Project MAToS, December 12th and 13th, 2016, Toulouse, France (with Radu Ignat)
• Special Session Nonlinear waves in dispersive equations at the 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications, July 1st to 5th, 2016, Orlando, Florida, USA (with François Genoud)
• Analytical and Numerical Advances Around Schrödinger Equations, Toulouse, October 22th to 26th, 2012 (with Rémi Carles, Fanny Delebecque and Claudia Negulescu)
• Workshop "Nonlinear PDE day", Toulouse, May 30th, 2012
• Journée Schrödinger, Toulouse, January 27th 2012 (with Claudia Negulescu).
• Conference "Nonlinear Dispersive PDE & related topics", Paris, June 14th-16th 2011 (with Raphaël Côte and Philippe Gravejat)

Long term scientific visits

• September 2024-July 2025 at theUniversity of McMaster (collaboration with Dmitry Pelinovsky).
• September 2014-July 2015 at the PIMS - University of British Columbia (collaboration with Tai-Peng Tsai and Stephen Gustafson).
• January 2007-June 2007 at the Mathematical Institute of the JLU Giessen in the team of Thomas Bartsch.