The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse: Self-Focusing and Wave Collapse

362 Pages·1999·4.3 MB·English

by  Catherine Sulem, Pirre-Louis Sulem (eds.)

Applied Mathematical Sciences
Volume 139
Editors
J.E. Marsden L. Sirovich
Advisors
S. Antman J.K. Hale P. Holmes
T. Kambe J. Keller K. Kirchgassner
B.J. Matkowsky C.S. Peskin
Springer
New York
Berlin
Heidelberg
Barcelona
Hong Kong
London
Milan
Paris
Singapore
Tokyo
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed. 34. Kevorkiardcole: Perturbation Methods in
2. Sirovich: Techniques of Asymptotic Analysis. Applied Mathematics.
3. Hale: Theory of Functional Differential 35. Carr: Applications of Centre Manifold Theory.
Equations, 2nd ed. 36. Bengr.sson/GhiUKtlllin: Dynamic Meteorology:
4. Percus: Combinatorial Methods. Data Assimilation Methods.
5. von MisedFriedrichs: Fluid Dynamics. 37. Saperstone: Semidynamical Systems in Infinite
6. Freiberger/Grenander: A Short Course in Dimensional Spaces.
Computational Probability and Statistics. 38. Lichtenberg/Lieberman: Regular and Chaotic
7. Pipkin: Lectures on Viscoelasticity Theory. Dynamics, 2nd ed.
8. Giacoglia: Perturbation Methods in Non-linear 39. PiccinUStampucchiaNidossich:O rdinary
Systems. Differential Equations in Rn.
9. Friedrick Spectral Theory of Operators in 40. Naylor/Sell: Linear Operator Theory in
Hilbert Space. Engineering and Science.
10. Stroud: Numerical Quadrature and Solution of 41. Sparrow: The Lorenz Equations: Bifurcations,
Ordinary Differential Equations. Chaos, and Strange Attractors.
11. Wolovich: Linear Multivariable Systems. 42. Guckenheimer/Holmes: Nonlinear Oscillations,
12. Berkovitz; Optimal Control Theory. Dynamical Systems, and Bifurcations of Vector
13. BlurnardCole: Similarity Methods for Fields.
Differential Equations. 43. OckendotvTaylor: lnviscid Fluid Flows.
14. Yoshizawu: Stability Theory and the Existence 44. Pazy: Semigroups of Linear Operators and
of Periodic Solution and Almost Periodic Applications to Partial Differential Equations.
Solutions. 45. Glashoff/Gustafson: Linear Operations and
15. Braun: Differential Equations and Their Approximation: An Introduction to the
Applications, 3rd ed. Theoretical Analysis and Numerical Treatment
16. Lefschetz; Applications of Algebraic Topology. of Semi-Infinite Programs.
17. CoNaffletferling: Optimization Problems. 46. Wilcox: Scattering Theory for Diffraction
18. &?~rIdt'rP: attern Synthesis: Lectures in Gratings.
Pattern Theory, Vol. I. 47. Hale et al: An Introduction to Infinite
19. MarsderdMcCracken: Hopf Bifurcation and Its Dimensional Dynamical Systems--Geometric
Applications. Theory.
20. Driver: Ordinary and Delay Differential 48. Murray: Asymptotic Analysis.
Equations. 49. Ladyzhenrkaya: The Boundary-Value Problems
21. Cnuranr/Friedrich.s: Supersonic Flow and Shock of Mathematical Physics.
Waves. 50. Wilcox: Sound Propagation in Stratified Fluids.
22. Rouche/Habets/Laloy: Stability Theory by 51. Golubitsky/Schaeffer: Bifurcation and Groups in
Liapunov's Direct Method. Bifurcation Theory, Vol. I.
23. Lamperti: Stochastic Processes: A Survey of the 52. Chipot: Variational Inequalities and Flow in
Mathematical Theory. Porous Media.
24. Grenunder: Pattern Analysis: Lectures in Pattern 53. Majdu: Compressible Fluid Flow and System of
Theory, Vol. 11. Conservation Laws in Several Space Variables.
25. Davies: Integral Transforms and Their 54. Wasow: Linear Turning Point Theory.
Applications, 2nd ed. 55. YO~idaO: perational Calculus: A Theory of
26. Kushner/Clark: Stochastic Approximation Hyperfunctions.
Methods for Constrained and Unconstrained 56. Chang/Howes Nonlinear Singular Perturbation
Systems. Phenomena: Theory and Applications.
27. de Boor: A Practical Guide to Splines. 57. Reinhardt: Analysis of Approximation Methods
28. Keihon: Markov Chain Models--Rarity and for Differential and Integral Equations.
Exponentiality. 58. Dwoyer/Hussaini/Voigt (eds): Theoretical
29. de Veuheke: A Course in Elasticity. Approaches to Turbulence.
30. Shiarycki: Geomelric Quantization and Quantum 59. SandersNerhulst: Averaging Methods in
Mechanics. Nonlinear Dynamical Systems.
31. Reid: Sturmian Theory for Ordinary Differential 60. Ghil/Childress: Topics in Geophysical
Equations. Dynamics: Atmospheric Dynamics, Dynamo
32. Meis/Markowitz: Numerical Solution of Partial Theory and Climate Dynamics.
Differential Equations.
33. Grenander: Regular Structures: Lectures in
Pattern Theory. Vol. tlI. (continued follr~wing index)
Catherine Sulem Pierre-Louis Sulem
The Nonlinear
Schrodinger
Equation
Self-Focusing and Wave Collapse
Springer
Catherine Sulem Pierre-Louis Sulem
Department of Mathematics CNRS UMR 6529
University of Toronto Observatoire de la C6te d'Azur
Toronto, Ontario, MSS 3G3 Bd. de l'observatoire, BP 4229
Canada 06304 Nice Cedex 4
[email protected] France
[email protected]
Editors
J.E. Marsden L. Sirovich
Control and Dynamical Systems, 107-81 Division of Applied Mathematics
California Institute of Technology Brown University
Pasadena, CA 9 1 125 Providence, RI 02912
USA USA
Mathematics Subject Classification (1991): 35Q55, 76B15, 76D33, 78A60, 82D10
With 9 figures.
Library of Congress Cataloging-in-Publication Data
Sulem, C. (Catherine), 1955-
The nonlinear Schrodinger equation: self-focusing and wave collapse1
Catherine Sulem, Pierre-Louis Sulem.
p. cm. - (Applied mathematical sciences; 139)
Includes bibliographical references and index.
ISBN 0-387-98611 -1 (alk. paper)
1. Schrodinger equation. 2. Nonlinear theories. L Sulem, P.L.
11. Title. III. Series: Applied mathematical sciences (Springer-
Verlag New York Inc.); v. 139.
QC174.26.W28S85 1999
530.12'44~21 98-53840
O 1999 Springer-Verlag New York, Inc.
All rights resewed. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York,
NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use
in connection with any form of information storage and retrieval, electronic adaptation, computer
software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the
former are not especially identified, is not to be taken as a sign that such names, as understood by
the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 0-387-98611 -1 Springer-Verlag New York Berlin Heidelberg SPIN 10689424
Preface
Regardez les singularit´es :
il n’y a que c¸a qui compte.
Gaston Julia
The nonlinear Schro¨dinger (NLS) equation provides a canonical descrip-
tion for the envelope dynamics of a quasi-monochromatic plane wave (the
carryingwave)propagatinginaweaklynonlineardispersivemediumwhen
dissipative processes are negligible. On short times and small propagation
distances, the dynamics are linear, but cumulative nonlinear interactions
result in a significant modulation of the wave amplitude on large spatial
andtemporalscales.TheNLSequationexpresseshowthelineardispersion
relation is affected by the thickening of the spectral lines associated to the
modulationandtheresonantnonlinearinteractions.Inoptics,itcanalsobe
viewed as the extension to nonlinear media of the paraxial approximation,
extensively used for linear waves propagating in random media.
TheNLSequationassumesweaknonlinearitiesbutafinitedispersionat
the scale of the carrying wave, while in situations where both dispersion
and nonlinearities are equally weak, a “reductive perturbative expan-
sion” leads to long-wavelength equations, like the Korteweg–de Vries, the
Benjamin–Onoor,inseveraldimensions,theKadomtsev–Petviashviliequa-
tions (Segur 1978, Ablowitz and Segur 1981). This class of equations also
includes the so-called derivative nonlinear Schr¨odinger (DNLS) equation
obeyedbydispersiveAlfv´enwavespropagatingalonganambientmagnetic
field in a quasi-neutral plasma, because of a phase-velocity degeneracy in
the dispersionless limit (see Mjølhus and Hada 1997 for a recent review).
The name “NLS equation” originates from a formal analogy with the
Schr¨odinger equation of quantum mechanics. In this context a nonlinear
potential (that is nonlocal in the Hartree description) arises in the “mean
vi Preface
field”descriptionofinteractingparticles.WhentheNLSequationisconsid-
eredinthewavecontext,thesecond-orderlinearoperator,whichdescribes
the dispersion and diffraction of the wave-packet, is however not necessar-
ily elliptic, and the nonlinearity arises from the sensitivity of the refractive
index to the medium on the wave amplitude.
Special attention is paid in this monograph to the “elliptic” NLS equa-
tion, which when written in a frame moving at the group velocity of the
carrying wave takes the simple form
∂ψ
i +∆ψ+g|ψ|2ψ =0,
∂t
with an attractive (g = 1) or repulsive (g = −1) nonlinearity. This equa-
tion together with its generalization to arbitrary power-law nonlinearities
g|ψ|2σψ is addressed in the three first parts of the monograph. The two
other parts deal respectively with situations involving a coupling to a
mean field that adiabatically follows the carrying wave modulation, and
to low-frequency acoustic waves driven by the modulation.
AfirstquestionrelatedtotheNLSequationconcernsthelinearstability
of a solution that is uniform in space and oscillatory in time. At the level
of the original problem, this corresponds to the effect of a slow modula-
tion on a monochromatic wave whose frequency is slightly shifted by the
nonlinearity.ItisstraightforwardtoshowthattheellipticNLSequationis
“modulationally” unstable (Benjamin–Feir instability) when g = +1 and
stable when g = −1. Note that some authors restrict the definition of the
“modulational instability” to the case of perturbations in the direction of
propagation and refer to the “filamentation instability” when the pertur-
bationsareinthetransversedirections.Weshallnotfollowthisrestriction
here.
Thefilamentationinstabilityiseasilyinterpretedinthecontextofnonlin-
earoptics,whereintheusualsituations,thewavemodulationisessentially
time-independent. In the NLS equation, the variable t then refers to the
coordinate along the beam, and the Laplacian is taken relatively to the
directions transverse to the propagation. For g = +1, the refractive index
increaseswiththewaveintensity,leadingtoaconvergenceoflightraysfrom
neighborhood sites towards the region of higher amplitude. This in turn
further increases the refractive index and leads to more light convergence,
resulting in a self-focusing of the wave at this location.
The phenomenon of wave-packet contraction can also occur in a one-
dimensional setting. The nonlinear development of the modulational
instability depends, however, strongly on the space dimension. When the
modulationispurelyone-dimensional,itleadstotheformationofsolitonic
structuresresultingfromanexactbalancebetweenthedispersiveandnon-
lineareffects.Inhigherdimensions,incontrast,thenonlinearitydominates
when the initial conditions are large enough in a suitable norm, resulting
in a blowup of the wave amplitude, if additional physical effects like dissi-
Preface vii
pation do not intervene to arrest the process. Since a spatial contraction
of the wave packet takes place together with the amplitude blowup, the
phenomenon is often called wave collapse in the physical literature. This
is a basic mechanism to produce a transfer of energy from large to small
scales,thuspermittingdissipativeprocessestoactandtoheatthemedium,
withpossibledegradationofthematerialinthecaseofadielectric.Inplas-
mas, the collapsing structures, often called “collapsons,” will act as sinks
for the wave energy. This phenomenon competes with the more gradual
energy transfer to small scales resulting from resonant wave interactions
(wave turbulence).
Thisbookmainlyreviewsthephenomenonofwavecollapse,describedby
theNLSequationorbyitsgeneralizationsinvolvingcouplingtootherfields.
Severalapproaches,rangingfromrigorousmathematicalanalysistoformal
asymptoticexpansionsandnumericalsimulations,arepresented,mostlyin
the case of localized solutions vanishing at infinity. The important issue of
the elliptic NLS equation with repulsive nonlinearity for solutions keeping
a finite amplitude at infinity is only briefly mentioned. This equation also
appearsinthedescriptionofaBosecondensate,acontextwhereitisoften
called the Gross–Pitaevskii equation. It admits solutions in the form of
coherent structures like vortices that define states that can be excited in
superfluid helium.
The validity of the NLS equation breaks down near collapse where the
underlying assumptions of small amplitude and large-scale modulation
(compared to the frequency and the wavelength of the carrier) no longer
hold. The attention paid to the nature of the singularity is, however, not
motivatedonlybyinterestinthemathematicalpropertiesofafundamental
equation of nonlinear physics. The features of the singularity strongly af-
fectthephysicalprocesses,evenwhenthecollapseiseventuallyarrestedby
additional effects that become relevant after sufficiently small scales have
been formed.
Thebookincludesseveralpartsthatcovervariousaspectsoftheproblem,
inanattempttoputinperspectivetherigoroustheoryoftheNLSequation
and the physical understanding of the wave-collapse phenomenon.
Part I is an introduction to the physics of quasi-monochromatic waves.
Various equivalent derivations of the NLS equation used in the literature
are reviewed. An illustration is given in the context of optical waves in
a Kerr medium. In usual situations, the duration of a laser pulse is long
compared to the carrying wave period, and the modulation can be viewed
asstationary.Asalreadymentioned,theamplitudedynamicsarethengov-
ernedbythetwo-dimensionalNLSequation.Incontrast,fortheultrashort
pulses emitted by power lasers, “normal” or “anomalous” time dispersion
effects come into play. The Benjamin–Feir, or modulational, instability is
also reviewed, together with the existence of solitons in one space dimen-
sion and their instability relative to transverse perturbations. The formal
analogy between the NLS equation and the equations of hydrodynamics,
viii Preface
which is useful in the context of superfluids, is mentioned. The variational
formulation (Lagrangian and Hamiltonian) of the problem, which through
theNoethertheoremleadstoimportantconservationlaws,ispresentedto-
getherwiththe“varianceidentity,”whichisthebasicestimateforproving
finite-time blowup.
Part II is devoted to a survey of rigorous results on the NLS equation
mostly in the case of localized solutions vanishing at infinity. Existence
propertiesarestatedbriefly,withreferencetotheoriginalpapersortomore
mathematicallyorientedreviewsforthedetailsoftheproofs.Standing-wave
solutions(alsocalledsolitary waves)andtheirstability/instabilityarealso
discussed. Special attention is then paid to blowup solutions. Important
estimates leading to a quantitative characterization of the collapse are es-
tablishedindetail,inparticularatthe“criticaldimension”(dcr =2forthe
usual cubic NLS equation) where the phenomenon of L2-norm (or mass)
concentrationnearcollapsegivesarigorousbasistotheconceptof“strong
collapse” used in the physical literature.
Part III discusses numerical simulations of blowing up solutions and
presentsanasymptoticconstructionofcollapsingsolutions.Insupercritical
dimensions,theblowupisself-similar,andadetailedanalysisoftheprofile
of the solution is given. At critical dimension, such solutions are not possi-
ble,andself-similarityappearstobeweaklybroken.Adelicateasymptotic
analysisispresentedtocharacterizethecriticalblowup,aproblemthatre-
mainedachallengeformorethantwentyyearsbutisnowwellunderstood.
Theeffectsofperturbationsthatcanmodifyorevenarrestthecollapseare
alsoconsidered.Theyincludedissipation,normaldispersion,andsaturated
nonlinearities.
PartIVconsiderssituationswhereasinthewater-waveproblem,amean
field is driven by the amplitude modulation, as a result of quadratic non-
linearities in the primitive equations. The multiple-scale formalism is first
presentedforageneralnonlinearscalarwaveequationandthenillustrated
onafewmodelequations.Anexampleofadegeneratesituationisalsodis-
cussed.Thewater-waveproblem,whichpresentssometechnicaldifficulties
due to the integro-differential character of the primitive equations, is then
reviewedandtheaccuracyofthemodulationanalysisrigorouslyestimated.
TheresultingDavey–Stewartsonsystemgoverningthecouplingofthewave
amplitude with the mean field are analyzed in the various regimes which
establish according to the parameters of the medium.
Part V addresses situations where the wave modulation drives low-
frequency acoustic waves, which can strongly affect the collapse dynamics.
Several examples are presented. We first consider Langmuir oscillations,
which play a central role in the dynamics of a quasi-neutral plasma. Self-
focusing leads to the formation of collapsing density cavities where the
oscillating electric field is trapped and where dissipative processes produce
a local heating of the plasma. The Zakharov equations governing these
dynamics are derived and their properties analyzed. Recent mathemati-
Preface ix
cal results for the “scalar model” are reviewed. They include conditions
for blowup in a finite or infinite time and also properties of exact self-
similarsolutionsintwodimensions.Examplesofprogressivewavesarealso
considered for laser beams in plasmas and in the context of the filamen-
tation (transverse collapse) of Alfv´en waves propagating along an ambient
magnetic field. It turns out that this phenomenon is very sensitive to the
coupling to the magnetosonic waves that develop sharp fronts, creating
conditions for the onset of strong magnetohydrodynamic turbulence.
The present survey is by no means comprehensive, and we list here a
few reviews published in the field. A basic reference on nonlinear waves is
the book of Whitham (1974). Asymptotic methods for nonlinear waves in
various physical contexts are described in Infeld and Rowlands (1990). A
detailed discussion of the physical aspects of the self-focusing phenomenon
is found in Vlasov and Talanov (1997). Basic properties of the NLS equa-
tion are discussed by Rasmussen and Rypdal (1986). A detailed review
of the mathematical theory of the NLS equation is given by Cazenave
(1989, 1994), Strauss (1989), and Ginibre (1998) who concentrate on the
well-posedness of the Cauchy problem and on the long-time behavior of
global solutions. The asymptotic analysis of the collapse is reviewed in
Landman, LeMesurier, Papanicolaou, Sulem, and Sulem (1989) and more
recently in Sulem and Sulem (1997). A unified presentation of the effect
of perturbations on critical collapse is found in Fibich and Papanicolaou
(1997b). We refer to Newell and Moloney (1992) for an introduction to
nonlinear optics, and to Dyachenko, Newell, Pushkarev, and Zakharov
(1992)foradiscussionofthetwo-dimensionalcubicNLSequations,includ-
ing wave turbulence. A recent review of the phenomenon of self-trapping
of optical beams in self-focusing media is presented in Segev and Stege-
man(1998),whiletransverseinstabilitiesofsolitarywavesarediscussedin
Kivshar and Pelinovsky (1999). The physics of Langmuir collapse is exten-
sivelydiscussedinZakharov(1984).Otherreviewsonrelatedtopicsinclude
Thornhill and ter Haar (1978), Rudakov and Tsytovich (1978), ter Haar
andTsytovich(1981),Goldman(1984),Zakharov,Musher,andRubenchik
(1985), Robinson (1997), Berg´e (1998). Various aspects of the nonlinear
dynamics of dispersive waves are discussed in the conference proceedings
edited by Balabane, Lochak, and Sulem (1989).
Weconcludethisdescriptionbystressingthedynamicrelevanceofsingu-
larity formation. Julia’s sentence (quoted by L. Garding, T. Kotake and J.
Leray,Bull.Soc.Math.France,92,263,1964)thatweputinepigraph,ex-
pressesinasomewhatprovocativewaytheimportanceofaneffectwhichis
sometimesconsideredas“unphysical.”Singularityformationrevealsdrastic
changes in the magnitude and the typical scales of the solution, although
in realistic situations other couplings intervene near collapse and act as
small-scalemollifiers.Fromaphysicalpointofview,varioussituationsare,
in fact, possible. In some instances, illustrated by the self-focusing of a
laser beam, the blowup of NLS solutions reflects the occurrence of violent
x Preface
(althoughpossiblynonstrictlysingular)eventsattheleveloftheprimitive
equations,givingavaluablepracticalinteresttotheenvelopeformalism.In
other cases, the NLS singularities only reflect the transition from a weakly
to a moderately nonlinear regime. An interesting question then concerns
the necessity to return to the primitive equations or the possible existence
of an intermediate asymptotics valid beyond the NLS blowup.
ThepresentmonographgrewfromasuggestionbyGeorgePapanicolaou
to write a survey on the phenomenon of wave collapse as described by the
NLS equation. We acknowledge with gratitude the enlightening collabora-
tiononthenatureoftheblowupsingularitieswehavehadwithGeorgeand
his collaborators B. LeMesurier, M. Landman, and X.P. Wang, during our
visits at the Courant Institute. Our work with W. Craig laid the basis of
our description of the water-wave problem. Our discussion of the coupling
tolow-frequencyacoustictypewavesiscloselylinkedtojointworkwithT.
Passot, S. Champeaux, and A. Gazol on the propagation of Alfv´en waves
in a magnetized plasma. We thank all of them for these fruitful collabo-
rations that provided the backbone of this book. We are very grateful to
L. Berg´e, V.S. Buslaev, J. Coleman, V. Dougalis, G. Fibich, N. Koppel,
E. Kuznetsov, V. Malkin, D. McLaughlin, H. Nawa, A. Newell, D. Peli-
novsky, G. Ponce, J.J. Rasmussen, J.C. Saut, J. Shatah, I.M. Sigal, A.
Soffer, W. Strauss, M. Weinstein, J. Xin, and V.E. Zakharov for very use-
fuldiscussionsorcorrespondencefromwhichwehavebenefitedthroughout
theyears,orfordetailedcommentsandsuggestionsonearlierdraftsofthe
manuscript.Wethankallourcolleagueswhogaveuscopiesoftheirpapers
beforepublication.WearealsogratefultoAchiDosanjh,FrankMcGuckin,
and the editorial and production staff at Springer-Verlag (New-York) for
their amiable and efficient help in preparing the book for publication. We
shallmaintainalistoferrataandcorrectionsthatwillbeavailablethrough
the world wide web address http://www.obs-nice.fr/sulem/. This work
benefitedfrompartialsupportfromNSERCoperatinggrantOGP0046179.
Toronto C. Sulem
Nice P.L. Sulem
March 1999