Since my PhD with J.-M. Ghidaglia,
I've been working on dispersive equations and their application in
optics, plasma physics and water-waves.
The general question is the following one: how can we describe complex, nonlinear, high frequence propagation phenomena can be described using
asymptotic analysis? For optics, plasma physics and water-waves the answer is: WKB-theory. The solutions are decomposed using a phase and an amplitude
assuming that the amplitude vary slowly compared tp the phase. All the game consists :
i) to find a small parameter that has a physical signification in the orginal system,
ii) to be able to perform a multiscale asymptotic expansion in the suitable time and space scale,
iii) to keep the main feature of the physical situation.
The basic theory is now well understood and one obtains transport equations, nonlinear Schrödinger equations, Zakharov systems, Davey-Stewartson systems,
KdV equations.... but the justification of the convergence and of the stability of the expansions is for from beeing complete (the problems are still open for
Zakharov systems, Davey-Stewartson and Nonlinear Schrödinger in some cases!)
The physical situations are often more complicated and the reallity exhibits more complex situation and one has to keep coupling phenomena. It is especially usefull
for water-waves (counter-propagating waves), laser-plasma interaction (Raman amplification) and nolinear optics (ultra-short pulse). One has to build intermediate
model that are less complexe that the original systems like Maxwell or Euler but that are less caricatural that the simple enveloppe model. A challenge is to be able
to describe complexe physical situation, with a reasonable numerical cost. On of the main challenge is to produce 3D numerical simulations in real physical configurations.
My collaborators in this direction are G. Métivier, D. Lannes, J. Bona, C. Galusinski, G. Gallice, B. Nkonga, M. Colin.
Interaction of two solitons with KdV: Raman amplification (Colin-Colin):
The general question is the following one: how can we describe complex, nonlinear, high frequence propagation phenomena can be described using
asymptotic analysis? For optics, plasma physics and water-waves the answer is: WKB-theory. The solutions are decomposed using a phase and an amplitude
assuming that the amplitude vary slowly compared tp the phase. All the game consists :
i) to find a small parameter that has a physical signification in the orginal system,
ii) to be able to perform a multiscale asymptotic expansion in the suitable time and space scale,
iii) to keep the main feature of the physical situation.
The basic theory is now well understood and one obtains transport equations, nonlinear Schrödinger equations, Zakharov systems, Davey-Stewartson systems,
KdV equations.... but the justification of the convergence and of the stability of the expansions is for from beeing complete (the problems are still open for
Zakharov systems, Davey-Stewartson and Nonlinear Schrödinger in some cases!)
The physical situations are often more complicated and the reallity exhibits more complex situation and one has to keep coupling phenomena. It is especially usefull
for water-waves (counter-propagating waves), laser-plasma interaction (Raman amplification) and nolinear optics (ultra-short pulse). One has to build intermediate
model that are less complexe that the original systems like Maxwell or Euler but that are less caricatural that the simple enveloppe model. A challenge is to be able
to describe complexe physical situation, with a reasonable numerical cost. On of the main challenge is to produce 3D numerical simulations in real physical configurations.
My collaborators in this direction are G. Métivier, D. Lannes, J. Bona, C. Galusinski, G. Gallice, B. Nkonga, M. Colin.
Interaction of two solitons with KdV: Raman amplification (Colin-Colin):
1)
BKW theory and justification of models:
T. Colin et W. Ben Youssef, Rigorous derivation of Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, Mathematical Modelling and Numerical Analysis (M2AN), Vol. 34, No 4, 2000, pp.873-911. (preprint)
T. Colin et D. Lannes, Long wave-short wave resonnance in nonlinear geometric optics, Duke Math. J. vol. 107, N0 2, 351-419, 2001,
(preprint).
T. Colin, C. Galusinski, H.G. Kaper, Long waves in micromagnetism, Communications in PDE, 27 (2002), no. 7-8, 1625--1658, 3,
(preprint).
T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis 31(1) 2002, pp. 69-91, (preprint).
T. Colin and D. Lannes, Justification of and long-wave correction to
Davey-Stewartson systems from quadratic hyperbolic systems, Discret and continuous dynamical systems, vol.11, Number 1, 83-100, 2004 (preprint).
W. Ben Youssef and T. Colin, Rigorous derivation of the Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, proceedings of the fifth international conference on Mathematical and numerical aspects of wave propagation, St Jacques de Compostel, jully 2000, SIAM Proceedings.
T. Colin, G. Gallice, K. Laurioux, Intermediate models for laser beam propagation, SIAM Journal on Mathematical Analysis, vol 36, No5, pp. 1664-1688. (preprint)
T. Colin, G. Ebrard, G. Gallice and B. Texier, Justification of the Zakharov model from Klein-Gordon-wave systems, CPDE, vol. 29, No 9-10, 1365-1401, 2004. (preprint)
2) Numerical simulations:
T. Colin et P. Fabrie, Semidiscretization in time for Schrödinger-waves equations, Discret and continuous dynamical systems, vol4, No 4, 671-690, (1998). (preprint)
T. Colin, B. Nkonga, Numerical model for light interaction with two level atoms medium, Physica D, Vol. 188, No 1-2, p. 92-118, 2004,
(preprint).
T. Bouchères, A. Bourgeade, T. Colin, B. Nkonga and B. Texier, Study of a mathematical model for stimulated Raman scattering, Math. Models Methods Appl. Sci. 14 (2004), no. 2, 217--252. (preprint).
R. Abgrall, T. Colin, B. Nkonga, Etude du système de Schrödinger-Bloch modélisant la propagation d'un laser dans un gaz. Notes aux Compte-rendus de l'Acad. des Sciences, t. 333, Série I, 689-692, 2001.
T. Colin et B. Nkonga, Computing oscillatory waves of nonlinear hyperbolic systems using a phase-amplitude approach, proceedings of the fifth international conference on Mathematical and numerical aspects of wave propagation, St Jacques de Compostel, juillet 2000, SIAM Proceedings.
T. Colin and B. Nkonga, Multiscale numerical method for nonlinear Maxwell equations, DCDS B, Volume 5, Number: 3 , August 2005, Pages 631 -- 658 , (preprint).
T. Colin and V. Torri, Numerical scheme for the 2-D maxwell-Bloch equation modeling ultrashort pulses, (preprint).
R. Belaouar, T. Colin, G. Gallice, C. Galusinski, Numerical coupling of Landau damping and Raman amplification, (preprint).
T. Colin, G. Ebrard, G. Gallice, Semidiscretization in time for Nonlinear Zakharov Waves Equations, (preprint).
3) Some particular models:
T. Colin, On a nonlocal, nonlinear Schr\"odinger equation occuring in plasma
Physics, Nonlinear evolution equations and dynamical systems, NEED's 92, Edited by V. Makhankov, I. Puzynin and O. Pashaev, World Scientific, 1993.
T. Colin, Sur une équation de Schrödinger non linéaire et non locale, C. R. Acad. Sci. Paris, t. 314, Série I, No 6, pp. 449-452, 1992.
T. Colin, On the Cauchy problem for a nonlocal, nonlinear Schrödinger equation occuring in plasma Physics, Differential and Integral Equations, vol 6, Number 6, pp. 1431-1450, November 1993. (preprint)
T. Colin, On the standing waves solutions to a nonlocal, nonlinear Schrödinger equation occuring in plasma Physics, Physica D, 64, pp. 215-236, 1993. (preprint)
L. Bergé et T. Colin,
Un problème de perturbation singulière pour une équation d'enveloppe en physique des plasmas, Notes aux Comptes Rendus de l'Académie des Sciences, t320, Série 1, p31-34, 1995.
L. Bergé et T. Colin, A singular perturbation problem for an enveloppe equation in plasma Physics, Physica D 84, 437-459, 1995. (preprint)
L. Bergé, B. Bidégaray et T. Colin, A perturbative analysis of the time envelope approximation in strong Langmuir turbulence, Physica D vol. 95, 3-4, 351-379, 1996. (preprint)
T.Colin et M.I.Weinstein, On the ground states of vector nonlinear Schrödinger equations, Annales de l'Institut Henri Poincaré, Physique Théorique, vol. 65, 1, 57-79, 1996. (preprint)
T.Colin and A.Soyeur, Some singular limits for evolutionary Ginzburg-Landau equations, Asymptotic Analysis, vol. 13, 361-372, 1996.
(preprint).
M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential and Integral Equations, 17 (2004), no. 3-4, 297--330. (preprint).
M. Colin and T. Colin, A numerical model for the Raman Amplification for laser-plasma interaction, Journal of Computational and Applied Math. 193 (2006), no. 2, 535--562. (preprint).
M. Colin and T. Colin, Cauchy problem and numerical simulation for a quasi-linear Zakharov system describing laser-plasma interaction, Proceedings
of the conference on nonlinear analysis, 2004, Orlando, Nonlinear Analysis 63 (2005) e1679-e1686. (preprint).
R. Belaouar, T. Colin, G. Gallice, C. Galusinski Theoretical and numerical study of a quasilinear Zakharov system
describing Landau damping, M2AN vol. 40, No6, 961-986 (2007) (preprint)
R. Belaouard, T. Colin, G. Gallice, C. Gallusinski, Le problème
de Cauchy pour un système modèlisant l'amortissement Landau en physique des plasmas, séminaire X-EDP 2004, (preprint).
T. Colin, G. Métivier, Instabilities in Zakharov equations for laser propagation in a plasma, in Phase space analysis of PDEs, A. Bove, F. Colombini, D. Del Santo Ed., Progress in Nonlinear Differential equations and their Applications 69, Birkhäuser, 2006. (preprint)
R. Belaouard, T. Colin, G. Gallice, C. Galusinski, V. Tikhonchuk, Quasilinear electron acceleration in a driven plasma wave 31st EPS Conference on Plasma Phys. London, 28 June - 2 July 2004 ECA Vol.28G, P-5.047 (2004), (preprint).
R. Balaouard, T. Colin, G. Gallice, C. Galusinski, V. Tikhonchuk, Quasilinear electron acceleration in a driven plasma wave, Plasma Phys. Control. Fusion 49 (2007) 969–984. (preprint).
M. Colin, T. Colin, G. Métivier, Nonlinear models for laser-plasma interaction. Séminaire X-EDP 2006-2007. (Article)
4) General theory:
T. Colin, On the Cauchy problem for dispersive equations with nonlinear terms involving high derivatives and with arbitrarily large initial data, Nonlinear Analysis, Theory, Methods and Applications, vol 22, No7, 835-845, 1994. (preprint)
T. Colin, Smoothing effects for dispersive equations via a generalized Wigner transform, SIAM Journal for Mathematical Analysis, vol. 25, No6, 1622-1641, nov. 1994.
T. Colin, Effets régularisant pour des équations dispersives obtenus par une transformée de Wigner généralisée, C. R. Acad. Sci. Paris, t. 317, Série I, No 7, pp. 673-676, 1993.
5) Water-waves:
T.Colin, F. Dias and J.M.Ghidaglia , On rotational effects in weakly nonlinear water waves over finitedepth, European Journal of Mechanics, fluids, 14, No 6, 775-794, 1995. (preprint)
T.Colin, F. Dias and J.M.Ghidaglia, On modulation of weakly nonlinear water waves, Contempory Mathematics, AMS, vol. 200, 47-56, 1996. (preprint)
T. Colin et M. Gisclon, An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg-de Vries equation, Nonlinear Analysis TMA, vol. 46, No 6, Novembre 2001, 869-892. (preprint)
W. Ben Youssef and T. Colin, Rigorous derivation of the Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, proceedings of the fifth international conference on Mathematical and numerical aspects of wave propagation, St Jacques de Compostel, juillet 2000, SIAM Proceedings.
T. Colin et W. Ben Youssef, Rigorous derivation of Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, Mathematical Modelling and Numerical Analysis (M2AN), Vol. 34, No 4, 2000, pp.873-911. (preprint)
T. Colin et J.-M. Ghidaglia, Un problème mixte pour l'équation de Korteweg-de Vries sur un intervalle borné, Notes aux Compte-rendus de l'Acad. des Sciences, t. 324, Série I, 599-603, 1997.
T. Colin et J.-M. Ghidaglia, Un problème aux limites pour l'équation de Korteweg-de Vries sur un intervalle borné, actes des journées "Equations aux dérivées partielles" Saint-Jean de Monts, exposé No III, 1997.
T. Colin et J.-M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval, Advances in Differential Equations 6 (2001), no 12, 1463-1492. (preprint)
J. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, to appear in Archive for Rational Mechanics and Analysis, Volume 178 Number 3, 2005 (preprint.pdf).
Come back to the home page of T. Colin
T. Colin et W. Ben Youssef, Rigorous derivation of Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, Mathematical Modelling and Numerical Analysis (M2AN), Vol. 34, No 4, 2000, pp.873-911. (preprint)
T. Colin et D. Lannes, Long wave-short wave resonnance in nonlinear geometric optics, Duke Math. J. vol. 107, N0 2, 351-419, 2001,
(preprint).
T. Colin, C. Galusinski, H.G. Kaper, Long waves in micromagnetism, Communications in PDE, 27 (2002), no. 7-8, 1625--1658, 3,
(preprint).
T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymptotic Analysis 31(1) 2002, pp. 69-91, (preprint).
T. Colin and D. Lannes, Justification of and long-wave correction to
Davey-Stewartson systems from quadratic hyperbolic systems, Discret and continuous dynamical systems, vol.11, Number 1, 83-100, 2004 (preprint).
W. Ben Youssef and T. Colin, Rigorous derivation of the Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, proceedings of the fifth international conference on Mathematical and numerical aspects of wave propagation, St Jacques de Compostel, jully 2000, SIAM Proceedings.
T. Colin, G. Gallice, K. Laurioux, Intermediate models for laser beam propagation, SIAM Journal on Mathematical Analysis, vol 36, No5, pp. 1664-1688. (preprint)
T. Colin, G. Ebrard, G. Gallice and B. Texier, Justification of the Zakharov model from Klein-Gordon-wave systems, CPDE, vol. 29, No 9-10, 1365-1401, 2004. (preprint)
2) Numerical simulations:
T. Colin et P. Fabrie, Semidiscretization in time for Schrödinger-waves equations, Discret and continuous dynamical systems, vol4, No 4, 671-690, (1998). (preprint)
T. Colin, B. Nkonga, Numerical model for light interaction with two level atoms medium, Physica D, Vol. 188, No 1-2, p. 92-118, 2004,
(preprint).
T. Bouchères, A. Bourgeade, T. Colin, B. Nkonga and B. Texier, Study of a mathematical model for stimulated Raman scattering, Math. Models Methods Appl. Sci. 14 (2004), no. 2, 217--252. (preprint).
R. Abgrall, T. Colin, B. Nkonga, Etude du système de Schrödinger-Bloch modélisant la propagation d'un laser dans un gaz. Notes aux Compte-rendus de l'Acad. des Sciences, t. 333, Série I, 689-692, 2001.
T. Colin et B. Nkonga, Computing oscillatory waves of nonlinear hyperbolic systems using a phase-amplitude approach, proceedings of the fifth international conference on Mathematical and numerical aspects of wave propagation, St Jacques de Compostel, juillet 2000, SIAM Proceedings.
T. Colin and B. Nkonga, Multiscale numerical method for nonlinear Maxwell equations, DCDS B, Volume 5, Number: 3 , August 2005, Pages 631 -- 658 , (preprint).
T. Colin and V. Torri, Numerical scheme for the 2-D maxwell-Bloch equation modeling ultrashort pulses, (preprint).
R. Belaouar, T. Colin, G. Gallice, C. Galusinski, Numerical coupling of Landau damping and Raman amplification, (preprint).
T. Colin, G. Ebrard, G. Gallice, Semidiscretization in time for Nonlinear Zakharov Waves Equations, (preprint).
3) Some particular models:
T. Colin, On a nonlocal, nonlinear Schr\"odinger equation occuring in plasma
Physics, Nonlinear evolution equations and dynamical systems, NEED's 92, Edited by V. Makhankov, I. Puzynin and O. Pashaev, World Scientific, 1993.
T. Colin, Sur une équation de Schrödinger non linéaire et non locale, C. R. Acad. Sci. Paris, t. 314, Série I, No 6, pp. 449-452, 1992.
T. Colin, On the Cauchy problem for a nonlocal, nonlinear Schrödinger equation occuring in plasma Physics, Differential and Integral Equations, vol 6, Number 6, pp. 1431-1450, November 1993. (preprint)
T. Colin, On the standing waves solutions to a nonlocal, nonlinear Schrödinger equation occuring in plasma Physics, Physica D, 64, pp. 215-236, 1993. (preprint)
L. Bergé et T. Colin,
Un problème de perturbation singulière pour une équation d'enveloppe en physique des plasmas, Notes aux Comptes Rendus de l'Académie des Sciences, t320, Série 1, p31-34, 1995.
L. Bergé et T. Colin, A singular perturbation problem for an enveloppe equation in plasma Physics, Physica D 84, 437-459, 1995. (preprint)
L. Bergé, B. Bidégaray et T. Colin, A perturbative analysis of the time envelope approximation in strong Langmuir turbulence, Physica D vol. 95, 3-4, 351-379, 1996. (preprint)
T.Colin et M.I.Weinstein, On the ground states of vector nonlinear Schrödinger equations, Annales de l'Institut Henri Poincaré, Physique Théorique, vol. 65, 1, 57-79, 1996. (preprint)
T.Colin and A.Soyeur, Some singular limits for evolutionary Ginzburg-Landau equations, Asymptotic Analysis, vol. 13, 361-372, 1996.
(preprint).
M. Colin and T. Colin, On a quasilinear Zakharov system describing laser-plasma interactions, Differential and Integral Equations, 17 (2004), no. 3-4, 297--330. (preprint).
M. Colin and T. Colin, A numerical model for the Raman Amplification for laser-plasma interaction, Journal of Computational and Applied Math. 193 (2006), no. 2, 535--562. (preprint).
M. Colin and T. Colin, Cauchy problem and numerical simulation for a quasi-linear Zakharov system describing laser-plasma interaction, Proceedings
of the conference on nonlinear analysis, 2004, Orlando, Nonlinear Analysis 63 (2005) e1679-e1686. (preprint).
R. Belaouar, T. Colin, G. Gallice, C. Galusinski Theoretical and numerical study of a quasilinear Zakharov system
describing Landau damping, M2AN vol. 40, No6, 961-986 (2007) (preprint)
R. Belaouard, T. Colin, G. Gallice, C. Gallusinski, Le problème
de Cauchy pour un système modèlisant l'amortissement Landau en physique des plasmas, séminaire X-EDP 2004, (preprint).
T. Colin, G. Métivier, Instabilities in Zakharov equations for laser propagation in a plasma, in Phase space analysis of PDEs, A. Bove, F. Colombini, D. Del Santo Ed., Progress in Nonlinear Differential equations and their Applications 69, Birkhäuser, 2006. (preprint)
R. Belaouard, T. Colin, G. Gallice, C. Galusinski, V. Tikhonchuk, Quasilinear electron acceleration in a driven plasma wave 31st EPS Conference on Plasma Phys. London, 28 June - 2 July 2004 ECA Vol.28G, P-5.047 (2004), (preprint).
R. Balaouard, T. Colin, G. Gallice, C. Galusinski, V. Tikhonchuk, Quasilinear electron acceleration in a driven plasma wave, Plasma Phys. Control. Fusion 49 (2007) 969–984. (preprint).
M. Colin, T. Colin, G. Métivier, Nonlinear models for laser-plasma interaction. Séminaire X-EDP 2006-2007. (Article)
4) General theory:
T. Colin, On the Cauchy problem for dispersive equations with nonlinear terms involving high derivatives and with arbitrarily large initial data, Nonlinear Analysis, Theory, Methods and Applications, vol 22, No7, 835-845, 1994. (preprint)
T. Colin, Smoothing effects for dispersive equations via a generalized Wigner transform, SIAM Journal for Mathematical Analysis, vol. 25, No6, 1622-1641, nov. 1994.
T. Colin, Effets régularisant pour des équations dispersives obtenus par une transformée de Wigner généralisée, C. R. Acad. Sci. Paris, t. 317, Série I, No 7, pp. 673-676, 1993.
5) Water-waves:
T.Colin, F. Dias and J.M.Ghidaglia , On rotational effects in weakly nonlinear water waves over finitedepth, European Journal of Mechanics, fluids, 14, No 6, 775-794, 1995. (preprint)
T.Colin, F. Dias and J.M.Ghidaglia, On modulation of weakly nonlinear water waves, Contempory Mathematics, AMS, vol. 200, 47-56, 1996. (preprint)
T. Colin et M. Gisclon, An initial-boundary-value problem that approximate the quarter-plane problem for the Korteweg-de Vries equation, Nonlinear Analysis TMA, vol. 46, No 6, Novembre 2001, 869-892. (preprint)
W. Ben Youssef and T. Colin, Rigorous derivation of the Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, proceedings of the fifth international conference on Mathematical and numerical aspects of wave propagation, St Jacques de Compostel, juillet 2000, SIAM Proceedings.
T. Colin et W. Ben Youssef, Rigorous derivation of Korteweg-de Vries type systems from a general class of nonlinear hyperbolic systems, Mathematical Modelling and Numerical Analysis (M2AN), Vol. 34, No 4, 2000, pp.873-911. (preprint)
T. Colin et J.-M. Ghidaglia, Un problème mixte pour l'équation de Korteweg-de Vries sur un intervalle borné, Notes aux Compte-rendus de l'Acad. des Sciences, t. 324, Série I, 599-603, 1997.
T. Colin et J.-M. Ghidaglia, Un problème aux limites pour l'équation de Korteweg-de Vries sur un intervalle borné, actes des journées "Equations aux dérivées partielles" Saint-Jean de Monts, exposé No III, 1997.
T. Colin et J.-M. Ghidaglia, An initial-boundary-value problem for the Korteweg-de Vries equation posed on a finite interval, Advances in Differential Equations 6 (2001), no 12, 1463-1492. (preprint)
J. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, to appear in Archive for Rational Mechanics and Analysis, Volume 178 Number 3, 2005 (preprint.pdf).
Come back to the home page of T. Colin