JONES, CHRISTOPHER / Mathematics / Researchers

International Center for Scientific Research

Researchers

Mathematics / JONES, CHRISTOPHER

Ckrtjcfm.brown.edu

Position

Professor at the Division of Applied Mathematics, and Director of the Lefschetz Center For Dynamical Systems, Brown University, USA.

Research interests

Dynamical systems as a tool for solving problems which originate in applications is the main thrust of Jones's research. He has been interested in the dynamics of optical systems, resonance phenomena in celestial mechanics and pattern formation. A particular focus has been the use of dynamical systems methods in the study of nonlinear wave motion. He has developed techniques for determining the stability of traveling and standing waves with applications in nerve impulse and optical beam propagation. He has recently worked on the problem of stable (multiple) pulse propagation in optical fibers with phase-sensitive amplifiers. He has also been involved in a group effort to understand transport and mixing in mesoscale ocean structures. He has focussed on the use of geometric methods, particularly invariant manifolds in phase space. A major collaborative mathematical study has been on the use of geometric methods in singularly perturbed systems.

Prizes and awards

Faculty Mentor of the Year Award, 1999, New England Board of Higher Education
Plenary Lecturer, SIAM Conference on Applications of Dynamical Systems, Snowbird, 1997
CNR Visting Professor and CIME Lecturer, Italy, June 1994
Humboldt Award for Senior U.S. Scientist, 1993-1994
Graduate Research Board Award, University of Maryland, 1989

Publications

''Spherically symmetric solutions of a reaction-diffusion equation'' Journal of Differential Equations, 49 (1983) 142-169

''Asymptotic behavior of a reaction-diffusion equation in higher space dimensions'' Rocky Mountain Journal of Math, 13 (1983) 355-364

''Characterization of bifurcation from the continuous spectrum by nodal properties'' Journal of Differential Equations, 45 (1984) 196-220 (with T.Küpper)

''Stability of the travelling wave solution of the FitzHugh-Nagumo system'' Transactions of the American Mathematical Society, 286 (1984) 431- 469

''On the infinitely many solutions of a semilinear elliptic equation'' SIAM Journal of Math Analysis, 17 (1986) 803-835 (with T.Küpper)

''Chaos and coherent structures in partial differential equations'' Proceedings of the Symposium on Coherent Structures, Physica D, 18 (1986) 85-112 (with A.Aceves et al.)

''Instability of standing waves in nonlinear optical waveguides'' Physics Letters A, 117 (1986) 175-180 (withJ.Moloney)

''Instability of standing waves for nonlinear Schrödinger type equations'' Ergodic Theory and Dynamical Systems, 8 (1988) 119-138

''An instability mechanism for radially symmetric standing waves of nonlinear Schrödinger equations'' Journal of Differential Equations, 17 (1988) 34-62

''A shooting argument with oscillation for semilinear elliptic radially symmetric equations'' Proceedings of the Royal Society of Edinburgh, 108A (1988) 165-180 (with T.Küpper and H.Plakties)

''Radial solutions of a semilinear elliptic equation at the critical exponent'' Archive for Rational Mechanics and Analysis, 104 (1988) 251-270

''Jumping stable manifolds for dissipative maps of the plane'' Physica D, 35 (1988) 87-106 (with S.Hammel)

''Invariant manifolds for semilinear partial differential equations'' Dynamics Reported, 2 (1989) 1-38 (with P.Bates)

''Travelling waves of a perturbed diffusion equation arising in a phase field model'' Indiana University Journal of Mathematics, 30 (1989) 1198-1222 (with R.Gardner)

''A topological invariant arising in the stability analysis of travelling waves'' Journal für die reine und angewandte Mathematik, 410 (1990) 167-212 (with J.Alexander and R.Gardner)

''Nearly real fronts in a Ginsburg-Landau equation'' Proceedings of the Royal Society of Edinburgh, 116A (1990) 193-206 (with T.Kapitula and J.Powell)

''A stability index for steady state solutions of boundary value problems for parabolic systems'' Journal of Differential Equations, 91 (1991) 181-203 (with R.Gardner)

''Stability of travelling wave solutions of diffusive predator-prey systems'' Transactions of the American Mathematical Society, 377 (1991) 465-524 (with R.Gardner)

''Competition between generic and non-generic fronts in envelope equations'' Physical Review A, 44 (1991) 3636-3652 (with J.Powell and A.Newell)

''Tracking invariant manifolds with differential forms in singularly perturbed systems'' Journal of Differential Equations, 108 (1994) 64-88 (with N.Kopell)

''A geometric proof of the Kwong-McLeod uniqueness result'' SIAM Journal of Math Analysis, 24 (1993) 436-443 (with C.Clemons)

''Existence and stability of asymptotically oscillatory double pulses,'' Journal für die reine und angewandte Mathematik, 446 (1994) 49-79 (with J.Alexander)

''Stability of travelling waves in non-convex scalar viscous conservation laws'' Communications in Pure and Applied Math,46 (1993) 505-526 (with R.Gardner and T.Kapitula)

''Existence and stability of asymptotically oscillatory triple pulses'' ZAMP, 44 (1993) 189-200 (with J.Alexander)

''Instability of quasiperiodic solutions of the Ginsburg - Landau equations'' Proceedings of the Royal Society of Edinburgh, 125A (1995) 501-517 (with A.Doelman and R.Gardner)

''A mechanism for capture into resonance'' Physica D, 69 (1993) 85-106 (with T. Burns)

''Tracking invariant manifolds up to exponentially small errors'' SIAM Journal of Math Analysis, 27 (1996) 558-577 (with T.Kaper and N.Kopell)

''Invariant Manifolds and Singularly Perturbed Boundary Value Problems'' SIAM Journal of Numerical Analysis, 31 (1994) 1558-1576 (with S.-K.Tin and N.Kopell)

''Stability of the in-phase travelling wave solution in a pair of coupld nerve fibers'' Indiana University Journal of Math, 44 (1995) 189-220 (with A.Bose)

''Settling and asymptotic motion of aerosol particles in a cellular flow field'' J. Nonlinear Science, 5 (1995) 337-358 (with M.Maxey and J.Rubin)

''The existence of travelling wave solutions of a generalized phase-field model'' SIAM Journal of Math Analysis, 28 (1997) 60-93 (with P.Bates, P.Fife and R.Gardner)

''Stability of pulses on optical fibers with phase-sensitive amplifiers'' ZAMP, 48 (1997) 175-192 (with J.Alexander, M.Grillakis, B.Sandstede)

''Existence and stability of N-pulses on optical fibers with phase-sensitive amplifiers'' Physica D, 106 (1997) 167-206 (with B.Sandstede and J.Alexander)

''Phase field models for hypercooled solidification'' Physica D, 104 (1997) 1-31 (with P.Bates, P.Fife and R.Gardner)

''Quantifying transport in numerically generated velocity fields'' Physica D, 110 (1997) 105-122 (with P.Miller, A.Rogerson and L.Pratt)

''Existence of standing pulse solutions to an inhomogeneous reaction-diffusion equation'' Journal of Dynmaics and Differential Equations,10 (1998) 1-35 (with J.Rubin)

''Bifurcations and edge oscillations in the semiconductor Fabry-Pé rot interferometer'' Optics Communications, 140 (1997) 93-98 (with J.Rubin)

''Viscous perturbations of vorticity conserving flows and splitting of separatrices'' Nonlinearity, 11 (1998) 47-77(with B.Sandstede and S.Balasuriya)

''The existence of travelling wave solutions of a generalized phase field model'' SIAM Journal of Math Analysis, 28 (1997) 60-93 (with P.Bates, P.Fife and R.Gardner)

''Average dynamics of the optical soliton in communication lines with dispersion management: analytical results'' Phys Rev E, 58 (1998) 48-51 (with A.Aceves, S.Turitsyn and V.Zharnitsky)

''Hamiltionian averaging in soliton-bearing systems with a periodically varying dispersion'' Phys Rev E, 59 (1999) 3843-3846 (with S.Turitsyn, A.Aceves and V.Zharnitsky)

''Lagrangian motion and fluid exchange in a barotropic meandering jet'' Journal of Physical Oceanography, 29 (1999) 2635-2655 (with A. Rogerson, P.Miller and L.Pratt)

''Bifurcation of internal solitary wave modes from the essential spectrum'' Phys Rev E, 61 (2000) 3114-3121 (with P.Kevrekidis)

''Bifurcation of asymmetric solutions in nonlinear optical media'' to appear in ZAMP (with T.Küpper and K.Schaffner)

''A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations'' Indiana U. Math. Journal, 49 (2000) 221-243 (with F.Geszetesy, Y.Latushkin and M.Stanislavova)

''Instability of multiple pulses in coupled nonlinear Schrödinger equations'' Phys Rev E,61 (2000) 5886-5892 (with A.C.Yew and B.Sandstede)

''Melnikov theory for finite-time vector fields'' Nonlinearity, 13 (2000) 1357-1377 (with B.Sandstede, S.Balasuriya and P.Miller)

''Exponentially small splitting of heteroclinic orbits: from the rapidly forced pendulum to discrete solitons'' Physics Letters A, 269 (2000) 120-129 (with P.Kevrekidis and T.Kapitula)

''Reconstructing basin-scale Eulerian velocity fields from simulated drifter data'' to appear in Journal of Physical Oceanography (with M.Toner, A.Poje. A.D.Kirwan Jr., B.Lipphardt and C.Grosch)

''Stabilizing effects of dispersion management'' to appear in Physica D (with V.Zharnitsky, E.Grenier and S.Turitsyn)

''Ground states of dispersion manged NLS'' Phys Rev E 62 (2000) 7358 (with V.Zharnitsky, E.Grenier, J.Hesthaven and S.Turitsyn)

''Stability of neuronal pulses composed of two concatenated unstable kinks'' to appear in Phys Rev E (with M.Romeo)

''Soliton internal modes:pure power law?'' Phys Rev E, 63 (2001) 036602 (with P.Kevrekidis and T.Kapitula)

''Diffusive draining and growth of eddies'' to appear in Nonlinear Processes in Geophysics (with S.Balasuriya)

''Chaotic transport of mass and potential vorticity for an island recirculation'' to appear in Journal of Physical Oceanography (with P.Miller, L.Pratt and K.Helfrich)

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