Table Of ContentSingularities in Fluids, Plasmas and Optics
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Series C: Mathematical and Physical Sciences - Vol. 404
Singularities in Fluids,
Plasmas and Optics
edited by
Rüssel E. Caflisch
Department of Mathematics,
University of California at Los Angeles,
Los Angeles, California, U.S.A.
and
George C. Papanicolaou
Courant Institute of Mathematical Sciences,
New York University,
New York, New York, U.S.A.
Springer Science+Business Media, B.V.
Proceedings of the NATO Advanced Research Workshop on
Singularities in Ruids, Plasmas and Optics
Heraklion, Greece
July 6-10,1992
Library of Congress Cataloging-in-Publicatio n Data
Singularitie s i n fluids , plasmas, and optic s : proceedings of a NATO
advanced researc h workshop Jul y 6-10, 1992, Heraklion , Greece /
edited by Russe] E. Caflisc h and George C. Papanicolaou.
p. cm. — (NATO ASI series . Serie s C, Mathematical and
physical sciences ; vol . 404>
ISBN 978-94-010-4894-1 ISBN 978-94-011-2022-7 (eBook))
DOI 10.1007/978-94-011-2022-7
1. Singularitie s (Mathematics)—Congresses. 2. Flui d mechanics-
-Mathematics—Congresses. 3. Plasma (Ionize d gases)—Mathematics-
-Congresses. 4. Optics—Mathematics—Congresses. 5. Mathematical
physics—Congresses. I . Caflisch , Rüsse l E. II . Papanicolaou,
George. III . Series : NATO ASI series . Serie s C, Mathematical and
physical sciences ; no. 403.
QC20.7.S54S65 1993
530.1'5—dc20 93-1726
ISBN 978-94-010-4894-1
Printed on acid-free paper
All Rights Reserved
© 1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint of the hardcover 1st edition 1993
No part of the material protected by this copyright notice may be reproduced or
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This book contains the proceedings of a NATO Advanced Research Workshop held within the programme
of activities of the NATO Special Programme on Chaos, Order and Patterns as part of the activities of the
NATO Science Committee.
Other books previously published as a result of the activities of the Special Programme are:
ABRAHAM, N.B., ALBANO, A.M., PASSAMANTE, A. and RAPP, P.E. (Eds.) -Measures of Complexity and
Chaos (B208) 1990 Plenum Publishing Corporation. ISBN 0-306-43387-7.
BUSSE, F.H. and KRAMER, L. (Eds.) -Nonlinear Evolution of Spatio-Temporal Structures in Dissipative
Continuous Systems (B225) 1990 Plenum Publishing Corporation. ISBN 0-306-43603-5.
CHARMET, J.e., ROUX, S. and GUYON, E. (Eds.) -Disorder and Fracture (B235) 1991 Plenum Publishing
Corporation. ISBN 0-306-43688-4.
MARESCHAL, M. (Ed.) - Microscopic Simulations of Complex Flows (B236) 1991 Plenum Publishing
Corporation. ISBN 0-306-43687-6.
MacDONALD, GJ. and SERTORIO, L. (Eds.) -Global Climate and Ecosystems Change (B240) 1991 Plenum
Publishing Corporation. ISBN 0-306-43715-5.
CHRISTIANSEN, P.L. and SCOTT, A.c. (Eds.) -Davydov's Soliton Revisted: Self-Trapping of Vibrational
Energy in Protein (B243) 1991 Plenum Publishing Corporation. ISBN 0-306-43734-1
HOLDEN, A.V., MARKUS, M. and OTHMER, H.G. (Eds.) -Nonlinear Wave Processes in Excitable Media
(B244) 1991 Plenum Publishing Corporation. ISBN 0-306-43800-3.
LING-LIE CHAU, NAHM, W. (Eds.) -Differential Geometric Methods in Theoretical Physics: Physics and
Geometry (B245) 1991 Plenum Publishing Corporation. ISBN 0-306-43807-0.
ATMANSPACHER, H. and SCHEINGRABER, H. (Eds.) - Information Dynamics (B256) 1991 Plenum
Publishing Corporation. ISBN 0-306-43912-3.
BABLOYAN1Z, A. (Ed.) - Self-Organisation, Emerging Properties, and Learning (B260) 1991 Plenum
Publishing Corporation. ISBN 0-306-43930-1.
PELm, L. (Ed.) -Biologically Inspired Physics (B263) 1991 Plenum Publishing Corporation. ISBN 0-306-
44000-8.
BISHOP, A.R., POKROVSKY, V.L. and TOGNETTI, V. (Eds.) - Microscopic Aspect of Nonlinearity in
Condensed Matter (B264) 1992 Plenum Publishing Corporation. ISBN 0-306-44001-6.
JIMENEZ, J. (Ed.) - The Global Geometry of Turbulence: Impact of Nonlinear Dynamics (B268) 1992
Plenum Publishing Corporation. ISBN 0-306-44014-8.
MOSEKILDE. E. and MOSEKILDE, L. (Eds.) - Complexity, Chaos and Biological Evolution (B270) 1992
Plenum Publishing Corporation. ISBN 0-306-44026-1.
ROY, A.E. (Ed.) -Predictability, Stability and Chaos in N-Body Dynamical Systems (B272) 1992 Plenum
Publishing Corporation. ISBN 0-306-44034-2.
BEN AMAR, M., PELCE, P. and TABELING, P. (Eds.) -Growth and Form: Nonlinear Aspects (B276) 1992
Plenum Publishing Corporation. ISBN 0-306-44046-6.
JENA, P. . KHANNA, S.N. and RAO, B.K. (Eds.) -Physics and Chemistry of Finite Systems: From Clusters to
Crystals (C374) 1992 Kluwer Academic Publishers. ISBN 0-7923-1818-8
SEGUR, H. (Ed.) -Asymptotics Beyond All Orders (B284) 1992 Plenum Publishing Corporation. ISBN 0-
306-44112-8
DAVIS, S.H., HUPPERT, H.E., MULLER, U. and WORSTER, M.G. (Eds.) Interactive Dynamics of Convection
and Solidification (E219) 1992 Kluwer Academic Publishers. ISBN 0-7923-1910-9
BOUNTIS, T. (Ed.) Proton Transfer in Hydrogen-Bonded Systems (B291) 1992 Plenum Publishing
Corporation. ISBN 0-306-44216-7
MARESCHAL, M. and HOLIAN, B.L. (Eds.) Microscopic Simulations of Complex Hydrodynamic
Phenomena (B292) 1992 Plenum Publishing Corporation. ISBN 0-306-44226-4
BOUNTIS, T. (Ed.) Chaotic Dynamics: Theory and Practice (B298) 1992 Plenum Publishing Corporation.
ISBN 0-306-44247-7
TABLE OF CONTENTS
PREFACE ix
I. Singularities in Fluids
Well-posed Numerical Calculations for Free-Surface Flows 1
G. Baker and S. Tanveer
On the Well-Posedness of Two Fluid Interfacial Flows with Surface Tension 11
I.T. Beale, T.Y. Hou and I.S. Lowengrub
Can Three-Dimensional Ideal Flows Become Singular in a Finite Time? 39
M.E. Brachet, M. Meneguzzi, A. Vincent, H. Politano and P.L. Sulem
Singularity Formation for Models ofAxi-Symmetric Swirling Flow 53
R.E. Caflisch and P. Smereka
Complex Analytic Branching Structures in Porous Media Convection 65
T. Dombre
Singularities of the Velocity Field and Interaction Locality of Turbulence 75
G .E. Falkovich
Solitons, Euler's Equation, and the Geometry of Curve Motion 93
R.E. Goldstein and D.M. Petrich
Asymptotic and Numerical Results for Blowing-up Solutions to Semilinear
Heat Equations 111
I.B. Keller and I.S. Lowengrub
Scaling of a Singularity of Euler 131
R.M. Kerr
The Rayleigh Centrifugal Instability for Vortex Rings with Swirl 137
A. Lifschitz
Dynamics of Magnetic Vortex Rings 151
N. Papanicolaou
Morphology and Singularities of Vortex Flows 159
A. Pumir and E.D. Siggia
Topological Transitions in Hele-Shaw Flow 167
M.l. Shelley, R.E. Goldstein and A.I. Pesci
viii
ll. Singularities in Optics and Plasmas
Separatrices and Singularities 189
A. Bhattacharjee
Nonlinear Dynamics of Ultra-Cold Gas 197
V. Flambaum and E. Kuznetsov
The Interaction Representation for Nonlinear Schrodinger Equation Penleve
Problem 205
G.M. Fraiman
Self-Similar Regimes of Subsonic Langmuir Wave Collapse 213
V.M. Malkin and VN. Khudik
Asymptotic Profiles of Blow-up Solutions of the Nonlinear Schrodinger
Equation 221
H.Nawa
Modulational Stability of Periodic Solutions of the Kuramoto-Sivashinsky
Equation 255
D.T. Papageorgiou. G.C. Papanicolaou and Y.S. Smyrlis
Dynamic Rescaling for Tracking Point Singularities: Application to Nonlinear
SchrOdinger Equation and Related Problems 265
G. Papanicolaou, C. Sulem, P.L. Sulem and X.P. Wang
Dynamic Self-Focusing of Structured Laser Beams in Plasma 281
H.A. Rose and D.P. DuBois
Experimental Manifestation of Strong Langmuir Turbulence 301
A.M. Rubenchik:
On Long-Lived Singularities in the Nonlinear Shroedinger Equation 309
V.F. Shvets, N.E. Kosmatov and B.J. LeMesurier
On Collapsing Solutions of the Nonlinear Schroedinger Equation in
Supercritical Case 317
V.F. Shvets, N.E. Kosmatov and B.J. LeMesurier
Sharp Criteria for Wave Collapse and Wave Collapse Ideology in Application
to Optical Pulses Compression 323
S.K. Turitsyn
Fluid Analogies in Optics 333
C.O. Weiss, K. Staliunas, C.P. Smith. G. Slekys and M.F.H. Tarroja
PREFACE
The articles in this volume represent presentations at the NATO Ad
vanced Research Workshop (ARW) on Singularities in Fluids, Plasmas and
Optics, which was held at the Institute of Applied and Computational Math
ematics in Heraklion, Greece, on July 6-10,1992. The purpose of this Work
shop was to survey current knowledge on singularities in fluids, plasmas and
optics and to stimulate further research on the subject. The scope of the
talks included mathematical, numerical and physical aspects of singularities
and their applications.
Singularities or nearly singular behavior occur in a wide range of phys
ical and mathematical problems, such as nonlinear focusing of laser beams,
curvature singularities on vortex sheets and magnetic reconnection. As a co
herent and distinct phenomenon, a singularity can provide a relatively simple
solution that describes a complex physical state.
Topics discussed at this meeting included the following:
• Singularities for the nonlinear Schrodinger equation, as a model for the
propagation of a laser beam in a nonlinear optical material.
• Near blowup for solutions of the nonlinear Schrodinger equation with
saturation, or the Zakharov equations.
• Singularity formation on fluid interfaces and vortex sheets.
• Possible singularities in solutions of the Euler equations for 3D incom
pressible, inviscid flow.
• Computation of the formation and propagation of singularities in the
complex plane.
• Computational methods for singularities using adaptive rescaling.
The Workshop was attended by an international group of mathemati
cians, physicists and engineers. Two specific objectives of the meeting were
to stimulate the exchange of ideas between researchers in fluid mechanics and
those in optics and plasma physics, and to promote interactions between sci
entists from Russia and those from western countries.
Scientific and financial support for this Workshop was provided by the
NATO Science Committee as a part of their program on Chaos, Order and
ix
x
Patterns. We are sincerely grateful to the Science Committee and, in par
ticular, to David Crighton, who initially suggested an ARW on this topic.
It is a pleasure to also thank the Institute of Applied and Computational
Mathematics for its warm hospitality and stimulating environment. John
Papadakis, the Director of the Institute, and Renia Papanastasiou deserve
special thanks for their efforts in planning and running the Workshop.
Russel E. Caflisch
UCLA, Los Angeles
March 8, 1993
