Welcome to Claudio Muñoz's homepage
Departamento de Ingeniería Matemática,
Facultad de Ciencias Físicas y Matemáticas,
Universidad de Chile
Beauchef 851, Torre Norte, Piso 5,
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
Martel (Video) and Frank Merle (Video 1,
Master and PhD Students:
Jose M. Palacios (Master, defended August 2018), Christopher Maulen
(Ph.D. started March 2018), Maria E. Martinez (Ph.D. started June 2018).
Marcio Cavalcante (2017, U. Alagoas, Brazil), Chulkwang Kwak
(PUC, 2017-), Hanne Van den Bosch (CMM, 2018-2019).
Part of the editorial board of SN Partial Differential Equations and
Publications and preprints:
40. Decay in the one dimensional generalized Improved Boussinesq equation
(with Christopher Maulen), preprint
39. Soliton dynamics for the 1D NLKG equation with symmetry and in the absence of internal
modes (with M. Kowalczyk and Y. Martel), preprint
38. Asymptotic dynamics for the small data weakly dispersive one-dimensional hamiltonian ABCD system (with C.
Kwak), preprint arXiv:1902.00454.
37. Stability and instability of breathers in the U(1) Sasa-Satsuma and Nonlinear Schroedinger models (with M.
Alejo and L. Fanelli), preprint arXiv:1901.10381.
36. The Akhmediev breather is unstable (with M.A. Alejo and L.
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 to appear in IMRN.
34. On the asymptotic behavior of solutions to the Benjamin-Ono equation (with G.
Ponce), preprint arXiv:1810.02329. To appear in
33. The Calderon problem for quasilinear elliptic equations (with G. Uhlmann), preprint
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
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, Comm. Math. Phys. 361 (2018), no. 1, 127-141.
26. Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in
(with M.A. Alejo), preprint arXiv:1707.02595. PAMS 146 (2018), no. 5, 2225-2237.
25. Stability of KdV solitons on the half-line,
(with Marcio Cavalcante), preprint arXiv:1707.02608, to appear in Revista Mat. 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
(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
(with Anna Kazeykina),
arXiv:1603.06600, J. Diff. Eqns. 264 (2018), no. 7, 4822-4888.
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,
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),
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,
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.
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
7. The Gardner equation and the L^2-stability of the N-soliton solution of the Korteweg-de Vries
(with M.A. Alejo and L. Vega), arXiv:1012.5290, T.
of the AMS 365 (1),
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),
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
1. The two-dimensional Lazer-McKenna conjecture for an exponential non-linearity,
(with M. del Pino), arXiv:math/0608168, Journal
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. 1191412, Center
for Mathematical Modelling (CMM), and France-Chile ECOS-Sud C18E06.
Some videos about soliton dynamics:
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.
Currently I am an associate researcher at Center for Mathematical Modeling CMM (Conicyt PIA AFB170001), and
member of the ECOS Conicyt project France-Chile ECOS-Sud C18E06. I am also a member of Fondecyt Mathematics
Funding Board (2018-2020).
I was 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
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.
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
studies at DIM (2016-2018).
Multivariate Calculus (Fall 2018),
Advanced dispersive PDEs (Fall 2018).
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).