Welcome to Claudio Muñoz's homepage

Chargé de Recherche CNRS and Departamento de Ingeniería Matemática, Universidad de Chile

cmunoz@dim.uchile.cl


Postal Address:
Departamento de Ingeniería Matemática,
Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile
Beauchef 851, Torre Norte, Piso 5,
Santiago, Chile
Phone: (+56)(2) 2978 0737

Research interests: Nonlinear dispersive equations, in particular: collisions, well-posedness, stability and asymptotic stability and blow-up.

I obtained a Ph.D. in Mathematics at the University of Versailles Saint-Quentin, under the advise of Yvan Martel (Video) and Frank Merle (Video 1, Video 2).

Master and PhD Students: Jose M. Palacios (Master, defending August 2018), Christopher Maulen (Ph.D. started March 2018), Maria E. Martinez (Ph.D. started June 2018).
Postdocs: Marcio Cavalcante (2017, U. Alagoas, Brazil), Chulkwang Kwak (PUC, 2017-), Hanne Van den Bosch (CMM, 2018-).


Publications and preprints:

36. The Akhmediev breather is unstable (with M.A. Alejo and L. Fanelli), preprint Proc. Third Workshop on Nonlinear Dispersive Equations 2017 (Campinas, Brazil).
35. On the dynamics of zero-speed solutions for Camassa-Holm type equations (with M. Alejo, F. Cortez and C. Kwak), preprint arXiv:1810.09594.
34. On the asymptotic behavior of solutions to the Benjamin-Ono equation (with G. Ponce), preprint arXiv:1810.02329.
33. The Calderon problem for quasilinear elliptic equations (with G. Uhlmann), preprint arxiv:1806.09586.
32. Breathers and the dynamics of solutions to the KdV type equations (with G. Ponce), preprint arXiv:1803.05475, accepted in Comm. Math. Phys.
(improvement of a previous paper arXiv:1802.05547, not intended for publication)
31. Extended decay estimates for generalized BBM equations (with C. Kwak), preprint arXiv:1802.01925, to appear in Fields Inst. Comm.
30. Nonlinear stability of 2-solitons of the Sine-Gordon equation in the energy space (with J. M. Palacios), preprint arXiv:1801.09933, accepted in Ann. IHP C Analyse Nonlineaire.
29. The scattering problem for the abcd Boussinesq system in the energy space (with C. Kwak, F. Poblete and J. C. Pozo), preprint arXiv:1712.09256, to appear in J. Math. Pure Appl.
28. On asymptotic stability of nonlinear waves, (with M. Kowalczyk and Y. Martel), Laurent Schwartz seminar notes 2016-2017.
27. Scattering in the energy space for Boussinesq equations, (with F. Poblete and J.C. Pozo), preprint arXiv:1707.02616, to appear in Comm. Math. Phys.
26. Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear Electrodynamics, (with M.A. Alejo), preprint arXiv:1707.02595. Accepted in PAMS.
25. Stability of KdV solitons on the half-line, (with Marcio Cavalcante), preprint arXiv:1707.02608, to appear in Revista Math. Iberoam.
24. Instability in nonlinear Schrodinger breathers, arXiv:1608.08169, Proyecciones Vol. 36, no. 4 (2017) 653--683.
23. Nonexistence of small, odd breathers for a class of nonlinear wave equations, (with M. Kowalczyk and Y. Martel), arXiv:1607.06421, in L. Math. Phys. May 2017, Vol. 107, 5, pp. 921-931.
22. Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation. II. (with Anna Kazeykina), arXiv:1603.06600, to appear in J. Diff. Eqns.
21. Kink dynamics in the phi^4 model: asymptotic stability for odd perturbations in the energy space (with M. Kowalczyk and Y. Martel), arXiv:1506.07420, accepted in JAMS 30 (2017), 769-798.
20. Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation (with Anna Kazeykina), arXiv:1502.00968, J. Funct. Anal. 270 (2016), no. 5, 1744-1791.
19. Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons (with R. Cote, D. Pilod, and G. Simpson), arXiv:1406.3196, Arch. Rat. Mech. Anal. 220 (2016), no. 2, 639-710.
18. On the Variational Structure of Breather Solutions I (with M.A. Alejo and J.M. Palacios), arXiv:1309.0625 accepted in JMAA Vol. 453 2, 1111-1138.
17. On the Variational Structure of Breather Solutions II (with M.A. Alejo and J.M. Palacios), arXiv:1309.0625, accepted in EJDE Vol. 2017 (2017), No. 56, pp. 1-26.
16. The Gardner equation and the stability of multi-kink solutions of the mKdV equation, arXiv:1106.0648, Discrete and Continuos Dynamical Systems 36 (7), 3811-3843.
15. Stability of integrable and nonintegrable structures, arXiv:1510.03366, Advances in Differential Equations 19 (9/10), 947-996.
14. Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers (with M.A. Alejo), arXiv:1308.0998, Anal. and PDE. 8 (2015), no. 3, 629???674.
13. On approximate controllability and stabilization of generalized KdV solitons, arXiv:1206.2430, SIAM Journal on Control and Optimization 52 (1), 52-81.
12. Multi-solitons for nonlinear Klein-Gordon equations, arXiv:1210.7953, (with Raphael Cote), Forum of Mathematics, Sigma 2.
11. Nonlinear stability of mKdV breathers (with M.A. Alejo), arXiv:1206.3157, Communications in Mathematical Physics 324 (1), 233-262.
10. On nonlinear stability of mKdV breathers (with M.A. Alejo), arXiv:1206.3151, in J. Phys. A: Math. Theor. Vol 45, no. 43.
9. Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection, arXiv:1009.4905, SIAM Journal on Mathematical Analysis 44 (1), 1-60.
8. Sharp inelastic character of slowly varying NLS solitons, preprint arXiv:1202.5807, not intended for publication.
7. The Gardner equation and the L^2-stability of the N-soliton solution of the Korteweg-de Vries equation (with M.A. Alejo and L. Vega), arXiv:1012.5290, T. of the AMS 365 (1), 195-212.
6. Inelastic character of solitons of slowly varying gKdV equations, arXiv:1107.5328, Communications in Mathematical Physics 314 (3), 817-852.
5. On the soliton dynamics under slowly varying medium for NLS equations, arXiv:1002.1295, Mathematische Annalen 353 (3), 867-943.
4. On the inelastic 2-soliton collision for generalized KdV equations, arXiv:0903.1240, IMRN 2010 (9), 1624-1719.
3. Soliton dynamics for generalized gKdV equations in a slowly varying medium, arXiv:0912.4725, Analysis & PDE 4 (4), 573-638.
2. Singular limits for the bi-laplacian operator with exponential non-linearity in R^4 (with M. Musso and M. Clapp), arXiv:0709.2878, Annales de l'IHP Non Linear Analysis 25 (5), 1015-1041.
1. The two-dimensional Lazer-McKenna conjecture for an exponential non-linearity, (with M. del Pino), arXiv:math/0608168, Journal of Differential equations 231 (1), 108-134.

All my papers are in the ArXiv, and Mathscinet, done in collaboration with several kind researchers.
My research is partially supported by Fondecyt Regular No. 1150202, Center for Mathematical Modelling (CMM), and Nucleus Millenium Center CAPDE.

Some videos about soliton dynamics: KdV Video 1, quartic KdV Video 2 and the wobbling kink of Sine-Gordon Video 3. A periodic mKdV breather with nonzero mean: Video 1, Video 2, and Video 3.


Projects:

I'm an associate researcher in the CAPDE collaboration, lead by Manuel del Pino. We run a biweekly seminar: CAPDE Seminar 2017

I was also part of the organizing committee of the 2nd. Workshop on Nonlinear Evolution Equations in Valdivia, held at Universidad Austral de Chile, Valdivia, on December 12-16 2016.
Partially funded by ERC Blowdisol, CAPDE Nucleo Milenio, CMM and EEQUADD MathAmSud Collaboration.
I was also organizer of the PDE session at Somachi 2015.
I was also part of the organizing comitee at CLAM 2016 Nonlinear dispersive equations session, taking place in Colombia between July 11-15 2016.
I'm also part of the MathAmSud collaboration EEQUADD 2016-2017. Workshop MathAmSud EEQUADD, to be held at Universidad de Chile, Santiago, on December 4-7 2017.
Partially funded by CAPDE Nucleo Milenio, CMM and EEQUADD MathAmSud Collaboration.
I was Director of Doctoral studies at DIM (2016-2018).


Teaching:

Current teaching:

Multivariate Calculus (Fall 2018),
Advanced dispersive PDEs (Fall 2018).

Past teaching:

Vector Calculus (Spring 2017)
Dispersive PDEs (Spring 2017)
ODEs (Spring 2017)
Multivariate Calculus (Fall 2017),
Advanced dispersive PDEs (Fall 2017) Dispersive PDEs (Spring 2016)
Multivariate Calculus (Spring 2016)
ODEs (Spring 2016)
Advanced PDEs (Spring 2015) (Main documents, in Spanish: Midterm 1, Midterm 2, Final Exam, some additional lecture notes).
Multivariate Calculus (Fall 2015).

At the University of Chicago (2011-2013):
Multivariate Calculus II,
Introduction to Complex Analysis,
Honors Calculus I-II-III, and
Mathematical Methods for Physics.

At the University of Chile:
Ordinary Differential Equations (2011).