Spectral and Dynamical Stability of Nonlinear Waves

368 Pages·2013·3.443 MB·English

by  Todd Kapitula, Keith Promislow (auth.)

Applied Mathematical Sciences
Todd Kapitula
Keith Promislow
Spectral and
Dynamical
Stability of
Nonlinear Waves
Applied Mathematical Sciences
FoundingEditors
FritzJohn,JosephLaselleandLawrenceSirovich
Volume 185
Editors
S.SAntman
DepartmentofMathematics
and
InstituteforPhysicalScienceandTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
[email protected]
P.Holmes
DepartmentofMechanicalandAerospaceEngineering
PrincetonUniversity
215FineHall
Princeton,NJ08544
[email protected]
K.Sreenivasan
DepartmentofPhysics
NewYorkUniversity
70WashingtonSquareSouth
NewYorkCity,NY10012
[email protected]
Advisors
L.Greengard
J.Keener
R.V.Kohn
B.Matkowsky
R.Pego
C.Peskin
A.Singer
A.Stevens
A.Stuart
Forfurthervolumes:
http://www.springer.com/series/34

Todd Kapitula • Keith Promislow
Spectral and Dynamical
Stability of Nonlinear Waves
123
ToddKapitula KeithPromislow
DepartmentofMathematics DepartmentofMathematics
andStatistics MichiganStateUniversity
CalvinCollege EastLansing,Michigan
GrandRapids,Michigan USA
USA
ISSN0066-5452
ISBN978-1-4614-6994-0 ISBN978-1-4614-6995-7(eBook)
DOI10.1007/978-1-4614-6995-7
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013934712
Mathematics Subject Classification (2010): 34Lxx, 35Pxx, 35Qxx, 37Kxx, 37Lxx, 47B15, 47B25,
47Dxx,47Exx,47Fxx,70Hxx,70Kxx
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Todd Kapitula: I am grateful tomy
colleagues, bothhere at thecollegeand
elsewhere,for theirsupport and
encouragement during thisendeavor. I am
especiallythankfulto my wife, Laura, for
herunwavering support during the
seeminglyendless rounds of revisions.
Finally,I wish tothankKeith for his
boundless enthusiasmfor, and perseverance
with,thisproject.
KeithPromislow: I thank Ronay, Sylvia,
Isaac, and Levi for tolerating my absence
during thewriting of thisbook, and Todd
fortolerating my presence. I amgrateful to
Roger Temamfor teaching methevalue of
writing books.

Foreword
Waves,patterns,andotherpermanentstructureshavealwaysplayedapiv-
otal role in applied mathematics. It is the proverbial no-brainer that they
should, and also that their stability to perturbations is an issue of impor-
tance. This is the first full-scale book devoted to the stability theory, de-
veloped over the past few decades, which lies at the interface of dynami-
calsystemsandfunctionalanalysis.Whilefunctionalanalysisprovidesthe
frameworkandaffordstheproperposingofquestions, itisdynamicalsys-
temsthatgivemanyoftheanswers.
TheEvansfunctionprovidesthekeybridgebetween thesetwotheoreti-
calviewpoints,andmuchofthistextconcernsitsdevelopment.JohnEvans
wrote his seminal papers in the 1970s. He had trained as a medical doc-
tor,butbecameenamoredwithmathematicsjustashewasabouttoembark
onhismedicalcareer,atleastatthepointwherehisfinancialoutlookwould
haveimproveddramatically.HegaveitalluptopursueaPhDinmathemat-
ics, but his mathematical interests still reflected his medical background.
Hodgkin–Huxley hadformulatedtheequations governingthepropagation
ofnerveimpulsesabout20yearsearlier,andhadcomputed,usingaprimi-
tivecalculator,travellingwavesthatrepresentedthepropagationofanerve
impulse. Itwas thediscovery ofthis wavethat showedthey had foundthe
mechanisms underlying the functioning of a nerve. So, it was known that
the wave was present in the equations, but surely it also had to be stable.
ShowingitsstabilitywasthetaskthatEvanssethimself.Hedidnotachieve
it, but he did build a theory that has had applications far beyond neuro-
science.
Whathappenednextwasindicativeofthefortuitouscircumstancesthat
are often involved in research advances. I had heard Evans speak in 1980
whileI wasa postdoctoralfellow inBritish Columbia,andrealized during
histalkthathisideasprovidedthemissingpiecetoaproblemthathadin-
terested many of us, namely the stability of the fast traveling pulse of the
FitzHugh–Nagumo equations. Although FitzHugh and Nagumo had sepa-
rately formulated this system as a simplification of the Hodgkin–Huxley
vii
viii Foreword
equations,Evanshimselfhadresistedlookingatthisproblemashewasaf-
teramoregeneralresult.Inhindsight,hadInotbeenluckyenoughtocome
across this connection, and thus been able to show the power of Evans’s
ideas,they mayneverhaveachievedtheprominenceandrangeofapplica-
tions they now enjoy. It took me most of the 1980s to convince the com-
munityoftheimportanceofwhatIhadbythennamedtheEvansfunction.
Gardner was working with me early on, and a major shift occurred when
weconvincedAlexandertojointheeffort;inparticular,histopologicaland
geometricexpertiseallowedustoputitinacompletelynewlight.Ishould
also mention Maginu in Japan who had realized early on the power of the
Evansfunction,andhisgroupmadeanumberofcontributions.
Simultaneously,thesubjectofstabilitywasgrowingupforproblemswith
Hamiltonianstructure.Toappliedmathematicians,thedivisionofphysical
problemsintoconservativeanddissipativeisfundamental,anditisnotsur-
prising that a stability theory evolved on both sides. Benjamin’s stability
resultforwaterwaveswasperhapsthefirstresultinthisarea,butmuchof
thestabilityworkonHamiltoniansystemswasdevelopedinRussia.Indeed,
itwasstimulatedbythefundamentalworkofArnol’dandZakharov’swork
onsolitonstability.Thisareaalsomaturedinthe1980s:Weinsteinresolved
thestabilityofstandingwavesofthenonlinearSchro¨dingerequationinhis
PhDthesis,andGrillakis,Strauss,andShatahbuiltacomprehensivetheory
forthestability ofwavesofalargeclassofHamiltonianpartialdifferential
equations.
The stability story was greatly enriched by the entwining of these two
threads.OfparticularnotewastheworkofPegoandWeinsteinwhoshowed
that the Evans function plays a key role in determining the stability of
the Korteweg–de Vries soliton. They showed that many of the calculations
carried out in conservative systems could be recast in terms of the Evans
function. This insight then led to the resolution of unsolved problems in
the stability of wave motionfor conservative partialdifferentialequations.
Zumbrunledaparalleldevelopmentinhisworkonthestabilityofviscous
shocks. This is an area marked by hard analytic estimates, but Zumbrun
managedtopushitmuchfurtherthroughtheuseofinsightsoriginatingin
theEvansfunctionapproach.
But that is all prehistory. This book is about how all this early research
grew into a full-blooded mathematical theory and how the importance of
thesestability methodscannowbemeasuredinalargenumberofapplied
areas. What began in nerve impulse equations has now had an impact in
nonlinearoptics,gasdynamics,fluidmechanics,materialsciences,combus-
tiontheory,andBose–Einsteincondensates,amongotherareas.
For agraduate student needing tolearn this material, a postdoc switch-
ing into an area that needs it, or even somebody (like me!) who claims to
be expert in this area, this book will be invaluable. It covers an enormous
amountofgroundanddoessowithgreatinsightandstyle.Ihaveoftenbeen
Foreword ix
toldthatIshouldhavewrittenthisbookandthatIshouldhavedoneitlong
ago!IamgladIdidnot,asKapitulaandPromislowhavedoneamuchbetter
jobthanIevercouldhave.
ChapelHill,NC ChristopherK.R.T.Jones