AD-A174 995 MiARKOVIAN SHOCK MODELS DETERIORATION PROCESSES i/ISTRATIFIED MARKOV PROCESSE (U) ARIZONA UNIV TUCSONAPPLIED MATHEMATICS PROGRAM A NEUELL 95 HAy "

UNCLASSIFIED RFOSR-TR -86-289 AFOSR-84-8256 F/G 2/ L

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MICROCOPY RESOLUTION TEST CHART

NATIONAL BUREAU OF ' IANDARDS 196 A

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AD- A 174 995*,u..,A ..

AD-A 74 F EPORT DOCUMENTATION PAGE............. 10. RESTRICTIVE MARKINGS

UTNCLASSIFIED2& SECURITY CLAUSIFICATION AUTHORITY 3. OISTRIBUTION/AVAILABILITY Of REPORT

Approved for public release; distribution2b. O&CLASSIF ICATION/OOWNGRAOING SCHEDULE unlimited.

4. PERFORMING ORGANIZATION REPORT NUMBERCS) S. MONITORING ORGANIZATION It goI'ORT fO,0

G.. NAME OF PERFORMING ORGANIZATION a, OFFICE SYMBOL 7s. NAME OF MONITORING ORGANIZATION

University of Arizona I fb" , Air Force Office of Scientific Research

6c. ADDRESS (Citi. Sate and ZIP Cod.j 7b. ADDRESS tCity. Stete and ZIP CO"deProgram in Applied Mathematics Directorate of Mathematical & Information

Mathematics Buflding #89 Sciences, Bolling AFB DC 20332Tllrcnn- Arizona 85721 ________________________

9& NAME OF PUNOINGAPONSORING .OFFICE SYMBOL 0. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBERORGANIZATION (if 4pek

AFOSR NM AFOSR-84-0256

k ADDRESS (city. State md ZIP Code) 10. SOURCE OF FUNDING NOB.PROGRAM PROJECT TASK WORK UNIT

Bolling AFB DC 20332 1 LEMENT NO. - NO. NO.

_61102F A511. TITLE fimelude Sewcurit egaaoujbAKOVIAN SHOCK MODELS, DETERIORATION PROCESSES, STRATIFIED MARKOV

PROCESSES AND REPLACEMENT POLICIES12. PERSONAL AUTHORCS)

Professor Alan Newell13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Yr.. Mo.. Day) ,S. PAGE COUNT

Final FROM 8/15/84 TOJA/5 1986, May, 5 161B. SUPPLIEMENTARY NOTATION

17 COSATI CODES SUBJECT TERMS (Continule on reuerse if neceirary and Identify by block number)

F91ELD GROUP Sue. GR. ~ L3,

19 ABSTRACT Coniinue on reverse if neceuari and idenhtf) by block numbert j ( ,'j " I

The funds from AFOSR-84-0256 were used to purchase a VAX 11/1750. A total of

33 reseach papers by 7 U. Arizona lathematicians were written with extensive

use being made of the VAX. On addition 2 Ph.D. students wrote doctoral

dissertations which Were heavily oriented toward scientific computation. -

20 OISTRIBUTION AVA ILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATIOhk~iNk'

UCLASSIFESC- %...IMITED Z SAME AS RPT - OTIC USERS 03 U" 1-D

22& NAME OF RESPONSIBLE INDIVIDUAL 22b, TELEPHONE NUMBER 22c. OFFICE SYMBOL

(Onelir Ame Codel

Dr. Arje Nachman ,(202, 767- 5028DO FORM 1473, 83 APR EDITION OF I JAN 73 IS OBSOLETE. rir:r:A5STFIFID

":.,ATY CLASS.F ATION OF THIS PAGO

.................................................................. "." %A.

This is an excellent record of achievment and a clear indication of the

benefit the AFOSR equipment can produce.

This report is certainly acceptable.

AFOSR.*Th. 8 6 - 2 189

Christooher Jones

"On the infinitely many solutions of' scmilire r elliptic uT. Kuepper), accepted by SIAM Jrnl. of Mathematical Analysis.

"Global dynamical behavior of the optical field in ring c-vity", 'with J.Hammel and J. Moloney), accepted by the Jrnl. of the Optical Society ofAmerica, Series 3.

"Chaos and coherent structures in parti:al differential oquitions", (with A.Aceves, H. Adachihara, J.C. Lerman, D. McLaughlin, J. Moloney and A.C.N ewell), Physica 1'D, 85-11? (19{{).

"Stabilty of' nonlinear waveguide modes", (with J. Volonry,, submlttrc toPhys. Rev. Letters.

"Invariant manifold theorems with aoplications" , to appear in Proc. NATOConference on Nonlinear Analysis, April, 1935, Italy.

"Invariant manifolds for semilincar partial differential equations",(with P. Bates), submitted to SIAM Review.

David 'IcL-ughlin

"A new c!aiss of instabilities in passive optical cavities", (with J. Kolonoyand A.C. Newell), Phys. Rev. Lett. 54, 691 (1985).

"Cnaos and Coherent Structures in Partial Differential Equations", (with A.Aceves et al.) Physica 18D, 85-112 (1986).

Jerome V. Moloney

"A new elass of instabilities in passive optical cavities", (with D.W.M'cLaughlin and A.C. Newell), Phys. Rev. Lett., 54, 631 (1935).

"Global dynamic=(l behavior of the optical fiell in a ring cavity", (with S.1 -7mme. rc, C. Jon.s), J. "ot. So'. Am. 3, Vol. I, % ;, (1985).

"Two dimensional transverse solit:ry waves as asymptotic states of the fieldin ,n oDti ,al r,,son-~'' -, E-i-.; ,j. 2u.mnt. !.l'>trorn., (1-21 , 1 302 (1885b).

"Plin. w : io moonltijonal inst-abilitis in m;ssiv. opticLl resonators",to . l. s;u i.Se 5v m i r--e P'u i ii- -omr ,.v, 1)',

Alan N'uw.l i

So i tonz :n - -. ' is .ne Paysi .i, , . :1 ', rAY V~~~1 ):: ).

"-e i,<ti a_) ''-s ro 0 so it{ rv s,',ve' i;, . '. n,'] T of" :' 'y' i '2 1 '. ,,th , w th

K nic ker bor rk,. r , . Fluid .'rch. , 15 1 1 ( :)

"Anr if'ini , di'nmnri r n roa o1t i-, bis T-Tili"v wf' :'fr'' I nd

c, aotic ;itt"rct ors contain sol i tary -4 ,V " 1 , (with J "oloney ind D.

86 12 11,....~~~ ~~~~ ~~~~ %I %... 1. _1 .. .. 1_ ... .. .... .... , .

'MiL,-u-h in), in 'hn-o s in !,onI in near Dynzimica 1 Sys toms, rTn. )4-1 19. 1oChand. SIAM (1984~).

"A n,-w c1~sof instabilities in passive optical civitits"1, (with D.'AcLaugnlin a-nd J. Moloney), Phys. Rev. Lott., 54, 681 (1985).

OCnjos and turbulence: is there a connection?", t~o appear in the SpecialPr c. of Confcr-nce on atetisApplied to Fluid 11 ,flhinics 'ind Sta-bilityciedica-ted in memory of Richard C. DiPrima, to be published by SIA>1,, 1986.

"T-, shvto- of's~tin dis Iocati,,ions", (wi th D. Iiror.),I Ph YSiCs Lett-rs ,Vol. 1 1 A , No. ,(18)

"Ir, losn ad ',oner'nt. structuares in Partial di ff, -rcnt iK oqEu:t1on:o (wi th A.- ,.oves o't -il) Phyvsica 1 8D, 95-11-1 (1986)

Two-limesp-it i-ic 4 otIl pattc-rns in ring, c-,vit n" , ( nt .YLuh nmJ1. MaIloney), to be submitted.

,17 hF i! Conditins" , with (Y. 7~n -,,z suntd I. o J. . Priynico

"Benrja-min-Feir turbulence in binary mixture, convect ion", Mwith Hi. Brand in(!P. n I,c ,pac'ar in Pnvsi'za P.

Gr'o'arv '3ak r

"S "nday Tn-'gc.1I~thods forAisntic:nTr'c-inin1 yl i-Tayloar instability Probloems" , (wi th 1). ',Pi ro n a!nd 3A . U r s z% P hys ic a 1L7 L,

.3oun"!,ry intral Tr' ( hniqu-,F for "ulti-ccan. cted DIoma..inf7", (wit,': '.3~.oicy),to apea,-r in J. Comp. Physics(16)

B ~ i Vmvlo £n~ i vof Fluid Lcr' n'' , (wi: a R.L. Vede',cy, .Vrd~r! --nd! A.. rszav), to i ppe:r in J. Fluid 'Ilech . (191-6).

"8 r'cf.&, r Fur'* on.-, :nc th" v ui- Probi-m" (wjithFl v'a' ~to in thef Proc. of' tho Royail Socioty of Lonnion(lh)

)"t' f !J,. ''' it mi I nstaibilt' (wt C. Ly"w ,a Ni s~ion For

Ava <'-

Dist

4 IX

Dainiel M eiron

"Goneraliz,-d !!ethods for Free S.'urfaice Flow Problems 11: Radiating Waves",(with G.R. Baker and S.A. Qrszag), in press.

":3ound-irv .7ntegral M-ethods for, Axisymmetri(e and Three-Dimensional Rayleigh-Taylor, Instability Problems", (with G.R. Baker and S.A. Orszag), Physica1 2D , 19-31 (193 4).

"The Sha3pe Of Stationarv DislocaItions", (with A.C. Newell), Physics Letters1 13a , 89lc ( 1935) .

"Difficulties with Three-Dimensional Weak Solutions for Inviscidincompressible Flow", (with P.G. Saffman), sub judice, Physics of Fluids,(191).! .

f..eiect ion of' Steady ,tates in the Two-Dimensionail Symmetric MIodel ofDendritic Growth", to appear, Phys. Rev. A, 33, (1936).

Michael Shelley

Ph.D. thosis: "Th.- Applicat-ion of Bounda-ry Integral Techniques in tMultiplyConnected Domains"

cedr Pcs.:rvngApproximati.ions t~o Successive Derivatives of Periodic* Functions by Iterated Splines", (with G. Baker), submitted for publication

to SiAM, jen. of' MurcricnlAaWi 18)

i-ltsuo Adcchihara

Thaos on en rue turee in Pairtial Di ff(crrnt i 1 quaitions" , (Wi th A.Aceves it l.), Physica 18D, 95-112 198)

4lejalndro Aceves

P 4 .r . isve i aP..c d t in tIhe on Api ec<'tmt s ec wr

on two tprob-! ens: trainping of nonlinear pulses on nunl inear waveguid cs byslitav-. Mh a aV a mp 1 itur irace-ndent r(-f' act i v" i nr-x; t-ur l us t: ,vi or-

which, .rises when declay affects are, included in the ikeda, ma,), this map)

r. ci ort- - i nt r

nc~ai- re no:pa,'a.-Ic.rr o f Iiti. ric na V 1t.

p 1 Ic ry r ui ~~ nw 0i i 3r a a c u''..c

n fo , , I ) j

ri:a -' - don-, how1rt.~ . a nv'sur

Il%

~r%

From this an-ysis, one has to solve a system of ODE's for' tneparameters of the soliton.

This has been done on the VAX. W . hive also integrated numrically thefull problem to see the effects of radiation.

All iuarzcddini

His work has been concentratcd on solving numerically the Ginzburg-Landauequation, sometimes referred to as the Newell-Whitehead equation, which hasmany physic.l a:plicatons. Tne equation is:

-"( " )W = X;i- (,Rr + i)W' W,

t r i xx

The boundary conditions are periodic.

Different parameter values are used to check the stability of the x-independent and the soliton solutions. The equation is integrated in timeusing a pseudo-spectral method. The linear part of the equation is solvedfor half time step using the FFT (the Fast Fourier Transform). Thenonlinear part is solved exactly for the remaining half-time step. Data is

collected and analyzed.

if

-v

I

bq2

-

Christopher Jones

Current Research

There are two major projects that form a focus for my research at

the moment. The first is a stability problem for travelling waves in a

fast-slow system of reaction-diffusion equations. The travelling wave is

constructed by piecing together solutions of reduced problems and the

question is to understand how the stability information for these reduced

problems is put together to determine the stability of the full problem.

This is related to my earlier work on the FitzHugh-Nagumo equations but

significantly more difficult as the underlying phase space dimension is

4 rather than 3. R. Gardner of the University of Massachusetts proved

the existence of the waves and the extra space dimension translates into

non-trivial behavior in the slow part of the system. The understanding

of the effect of this new feature on the stability problem is quite

subtle. Gardner and I are working on this problem jointly.

The second problem relates to standing wave solutions of the nonlinear

Schr~dinger equations. I have proved an instability result and applied

it to nonlinear optical waveguides with Moloney. I am currently involved

in extending the results to a wider class of waves. But what is more

interesting is that the results have suggested some striking connections

between the methods of proof used to find the waves in the first place,

namely, the dynamical systems approach (as developed by Kipper and myself)

and the variational approach (as develcped by Strauss and Berestycki-Lions).

These techniques have up until now developed independently and a relationship

between them will, I believe, open up the possibility of exciting new results.

.%4

Instability of Standing Waves for

Nonlinear Schridinger Type Equations

b y

Christopher K.R.T. Jones

Division of Applied MathematicsBrown University

Providence, Rhode Island 02912

Current Address:

Department of MathematicsUniversity of Maryland

College Park, Maryland 20742

V.

This work is dedicated to the memory of Charles Conley

This research was supported in part by the National Science Foundation NSFam Grants DMS 8501961 and DMS 8507056, Air Force Office of Scientific

Research AFOSR Grant #83-0027, and Army Research Office ARO ContractDAAG-29-83-K-0029.

-

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% % %

IInstability of Standing Waves in Nonlinear Optical Waveguides

C.K.R.T. Jones

Department of Mathematics

University of Arizona

% 'Tucson, Arizona 85721

U.S .A.

and

J.V. Moloney

Physics Department

He'iot-Watt University

Riccarton, Edinburgh EHI4 4AS

Scotland, U.K.

.4.

PACS numbers: 02.78 42.20

:4

Abstract

A new mathematical instability technique is presented and applied to

determine the stability properties of a physically important class of

standing waves in nonlinear planar optical waveguides. The method is

illustrated by a case where soliton perturbation techniques or variational

methods are inapplicable.

;& A

1

On the Infinitely Many Solutions of a Semilinear Elliptic Equation

/

C. Jones*

Department Mathematics

Uni rsity of Arizona

ucson, Arizona 85721

T. Kupper **

Abteilung der Mathematik

Universitat Dortmund

4600 Dortmund 50

West Germany

*Supported, in part, by the National Science Foundation under grants #I,CS

82U0392, INT8314095 and by the Air Force Office of Scientific Research grant

#AFOSR 83 3227.

*Supported in part by a grant from the Deutsche Forschungsgemeinschaft.

Lefschetz Center for Dynamical Systems

New Class of Instabilities in Passive Optical Cavities

D. W. McLaughlin, J. V. Moloney, and A. C. NewellApplied Mathematics Program and Optical Science C'nter, UniversitY. ol A r:ona, Tucson, A rizona 85721

(Received 2 July 1984)

In this Letter we show that the fixed points of the Ikeda map are more unstable to perturbationswith a short-scale transverse structure than to plane-wave perturbations. We correctly predict themost unstable wavelength, the critical intensity, and th~e growth rats ol these disturbances. Ourresult establishes that, for a large class of nonlinear waves, spatial structure is inevitable and drasti-cally alters the route to chaos. In an optical cavity the consequence is that the period-doubling cas-cade is an unlikely scenario for transition to optical chaos.

PACS numbers: 42.65.-k

In this Letter, we announce a new and unexpected in which diffraction effects are neglected. The purposeresult, an instability whose consequences have ramifi- of this Letter is to point out that this assumption is notcations for a large class of nonlinear wave problems justified even for cases in which the input beam is verywhose dynamics can be described by envelope equa- slowly varying in the transverse direction x(xy) andtions. Specifically, it deals with periodically forced the Fresnel number is large. The reason is that thefield equations of the universal nonlinear Schr6dinger fixed point solutions of the plane-wave map (1) aretype. This instability changes the whole character of more unstable to perturbations with a short-scalethe route of the system from a simple to a turbulent transverse structure than they are to perturbationsstate. It generates spatial structure, and the subse- with plane-wave structure. To emphasize this point,quent onset of chaotic behavior completely bypasses the numerical experiment discussed in this Letter isthe period-doubling scenario which is relevant if spa- run at a parameter value p for which the fixed pointstial structure is ignored. Moreover, the scenario which of the Ikeda map are stable!does emerge has a universal character of its own. Lx- This discover;7 has important ramifications. First, itamples of this phenomena are found in optics, either shows that the initial bifurcation of the system intro-in the transmission along optical fibers or in optically duces an extra dimension into the problem, a short-bistable cavities. t It is in the latter context that this wave transvCrsc excitation of temporal period two.work is presented. This extra dimension affects significantly the subse-

In this problem we are interested in the long-time quent behavior of the system. As the stress parame-state of a continuous laser signal which is recirculated ters are raised, no period-doubling cascades into chaosthrough a nonlinear medium. In examining one par-ticular manifestation of optical bistability (a ring cavitywith Kerr nonlinearity), Ikeda2 wrote a map expressing (a)the (complex) amplitude g, I of the electric field E D

on the (n + l)st pass through the cavity as a functionof electric field amplitude on the nth pass- 1I1

g.+ I = a + Rg,, exp[ i 0o + ipLN(1)/21. (1) BA

In (1), a is the amplitude of the input field, R < I the areflectivity of the mirrors, 00 the detuning parameter,p is (effectively) the length of the nonlinear medium,and N(gfg, ) measures its nonlinear response. Twocases are usually studied: (I) the saturable medium, °°' I 25

N(i) - (I +21)t (2) the Kerr medium, N(I) 00--------

=- I + 2!, which is the small intensity limit of the 075-- . saturable case. Equation (1), called the Ikeda map, is 50

a two-dimensional invertible map and exhibits a 025variety of behavior which is already well documentedin the literature.2-4 In various parameter ranges (the 0o 05 0 1.5 20two parameters which are varied are a and p), onefinds multiple fixed points (see Fig. 1) and sequences FIG. I. (a) The multivalued response of the amplitude ofof period-doubling bifurcations leading to chaotic at- the fixed point IgI vs a at fixed values of p for the Ikedatractors. map. (b) A graph of b(I.,r) vs f.fr=.,'-fKforu-pl pgg°

The map (1) invokes the plane-wave approximation equal to 0.11 and 0.24.

681

• - " ..... . . +.. , %. V % % 'd. "" _-.

v.b 07?

ability of nonlinear stationary waves guided by a thin film boundedby nonlin I ia

! Moloey. .

hyics Departm Ileriot Watt University Riccarton, Edinburgh EIIJ4 4AS. Scoil.cid-: UC T. Seaton, and G. I. StegemanOptical Sciences Center and Arizona Research Laboratories. University ofArizona. T':_son. - 'izona 85721

(Received 2 December 1985; accepted for publication 3 February 1986)

The stability of stationary, TED-type, nonlinear, thin-film guided waves was in, estigatednumerically for both symmetric and asymmetric planar waveguides with nonlinear cladding andsubstrate layers. It is found that large regions of the dispersion curves are unstable at high powers.

Unique properties' - 7 have been predicted for waves n2 [ W(x,z) 2 ] = n c, 1 W(xz)guided by thin films when one or more of the guiding media 1. ( (2a)exhibit a field-dependent refractive index. Self-focusing 22a)

bounding media lead to multiple new branches with power and

thresholds, as well as field distributions whose maxima shift n2 [I W(xz) 2) n: ( xl<d) . (2b)from the film to the bounding media with increasing pov- Substituting into the nonlinear wave equation and retaininger.'-' In fact, this geometry has been identified as an excel- only the first derivat: e of IV(xz) with respect to z leads tolent candidate for all-optical switching, with our withoutbistability. - To date, however, theoretical analysis has been 2ik,, +W(x) 0k2 : x,)based solely on steady-state solutions to a nonlinear wave 3z 3X2

equation which contains an intensity-dependent refractive X0 2 - n2 [Wxz)-Z]}W(x,z) = 0. (3)index. The salient question is whether these wave solutions Analytical stabil!ty analysis of Eq. (3) is complicated byare stable on propagation, and the consequences of possible the fact that it is a partial differential equation and, further-unstable regions to proposed devices. For the related prob- more, is a Hamilton:an system. The usual stability analysis

lem of self-focusing of plane waves in infinite media, Kolo- for dissipative dynamical systems does not apply (unless one

kolov has shown that the solutions are stable for dP/did> 0 deliberately introduces losses into the problem). One reasonwherePisthepowerandgisthepower-dependent refractive for the difficulty in siudving the stabilit. of Eq. (3) is thatindex. Numerical propagation studies of nonlinear waves many of the eigenvalues of the linearization of (3) lie on theguided by the interface between a self-focusing and a power- imaginary axis which is precisely the condition for instabilityindependent medium by Akhmediev and co-workers'"1 in a dissipative dynamical system. Although it has been pos-have led to a similar conclusion. Recently, a theory based on sible " to perform a s:abilit* analysis for TE,, with m = 0, it

phase portraits has been developed'' for the stability of TE,, is necessary to proceed numerically for tn =0. First, thewaves guided by a thin film bounded by self-focusing media, steady-state solutions ', ith W(x,z)--. W(x) were obtained inand the important conclusion of that work is that the waves the usual way' - to ortain a field solution corresponding to aare unstable on negatively sloped branches (dP d13 < 0) of particular pointto o ne of the nonlinear guided wave solu-

the nonlinear dispersion curve, and that they are stable on tion branches. This distribution was then assumed to be

positively sloped regions provided that self-focusing occurs launched at z = 0 and Eq. (3) was solved numerically for

in only one nonlinear medium. In this letter we report a test

of this conclusion via a numerical investigation of the stabil- Zity of TE,, solutions for films bounded by self-focusing me- ..dia. 0

The geometry analyzed consists of a film (ix! <d, re-fractive index n,, ) bounded by two nonlinear media with low 2d-

power indices n , and n:, as shown in Fig. 1. Since the numeri-cal analysis is performed in the slowly varying phase andamplitude approximation, we write the optical field as n,+aIEi z no n,

E(r) = k1W(xz)e " Oka - " + Cc. , (I)

where/ is the effective guided wave index. k, = 2,J/c and thevariation in the amplitude term W(x.:) along the propaga-tion direction z is assumed to be small over one wavelength.

The refractive index in the various media is given by FIG I Nonlinear Aa~ .ctry studied for .aae stabilitt

O2; ~ ~ ~ t 48,- ('( q 11 ) 3* Ma- td , .nQ, A' ,; Io. -: 01S01 0,0'::.Z .rCtt ''' . ' :S

' ,, , ot the Conference on MathematicsApplied to Fluid Mechanics and Stability Dedicated in Memory ofRichard C. DiPrima. Published by SIAM, December, 1985.

Chaos and Turbulence; is there a

connection?

by

Alan C. Newell*

Department of MathematicsUniversity of Arizona

Tucson, Arizona 85721

Abstract. In this essay we discuss the relation of chaos, which isthe unpredictable behavior associated with finite systems of ordinarydifferential and difference equations, and turbulence, which is theunpredictable behavior of solutions of infinite dimensional, nonlinearpartial differential equations. The evidence that there is someconnection, at least in certain regimes of parameter space, issufficiently convincing to provide the motivation to search foranalytical means for reducing the governing partial differentialequations to either a finite system of ordinary differential equationsor a much simpler partial differential equation of universal type. Asuccessful reduction scheme must capture the spatial structure of thedominant modes accurately and we suggest ways of finding thesestructures in certain limiting situations. Five such schemes arepresented and, in each case, the approximation is related in some wayto the presence of a small parameter, near critical, nearlyintegrable or nearly periodic. One of these reductions leads to thecomplex Ginzburg-Landau equation, which has universal character, andits importance is stressed. In connection with this equation, weintroduce the terms "wimpy" turbulence and "macho" turbulence toconnote the crucial differences between the behavior of its solutionin one and two space dimensions, a difference which has much in commonwith the contrast between two and three dimensional high Reynoldsnumber flows because of vorticity production. In the final section,several ideas concerning the nature of high Reynolds number, fullydeveloped turbulence are presented and the possible roles of singularsolutions and "fuzzy" attractors are discussed. Throughout the essay,we argue that before much new progress is made, one has to understandthe onset of spatial chaos, that is, the transition from a spatiallyregular state (possibly with a chaotic temporal behavior) to one inwhich the spatial power spectrum is broadband. This question is amajor focus of our present research program.

This contribution is dedicated to the memory of Dick DiPrima, a

good friend and long time colleague who left us too soon.

Supported by NSF-DMS-8403187, AFOSR - 83-0227, Army - DAAG-29-85-K091,ONR Physics - N00014-84-K-0420, ONR Engineering - N00014-85-K-0412.

i%

MRC Technical Summary Report #2830

BOUNDARY INTEGRAL TLCHINIQUES FOR

MULTI-CONNECTED DOMAINS

G. R. Baker and M. J. Shelley

Mathematics Research CenterUniversity of Wisconsin-Madison610 Walnut StreetMadison, Wisconsin 53705

June 1985

(iCuceived Mdrcil II, 1985)

Approved for public releaseDistribution unlimited

" : Sponsored by

National Aeronautics & Space Administration

U. S. Army Research Office Washington, DC 20546P. 0. Box 12211Research Triangle Park National Science FoundationNorth Carolina 27709 Washington, DC 20550

... , ,,.S-.5 -, , .-,-W,*,. . .. .

Phil. Tran). R. Soc. Lond. A 315, 405 422 , I9,5 405

Printed in Great Britain

The geometry of the Hill equation and of the Neumann system

By N. M. ERCOLANIt AND H. FLASCHKA

Department of Mathematics, The University of Arizona, Tucson, Arizona 85721, U.S.A.

Let there be given a finite:gap operator L = d2/dx 2 + q and its Baker function V1(x, p),which is analytic forp on a certain hyperelliptic curve C. It is shown that a sequenceof Backlund transformations maps C to a projective space. This embedding can beinterpreted as a matrix representation of the Hill equation by the Neumann systemof constrained harmonic oscillators. The image curve, C', lies on a rational ruledsurface; the structure of this surface is explained by use of ideas due to Burchnall &Chaundv (Proc. R. Soc. Lond. A 118, 557-583 (1928)). Baker functions and Bdcklundtransformations are then used to define a (many-to-many) correspondence betweeneffective divisors on the curve C and points lying on a quadric, or in the intersectionof two or more quadrics. This relates the theory of the Hill equation to earlier workof Knorrer, Moser and Reid. It is then shown that the Kummer image of theJacobianof C can be realized as a hypersurface in the space of momentum variables of theNeumann system. Further projects, such as extensions to non-hyperelliptic curves, areoutlined. -

, 1. INTRODUCTION

A differential operator L = D2 +q(x), 1) = d/dx, is said to be 'finite-gap' if it commutes with

a differential operator B of odd order, [L, B] = 0. A 'finite-gap potential' q(x) is therefbre atime-independent, or stationary, solution of an equation aL/ t = [B, L] in the Korteweg- deVries hierarchy. Because L and B commute, they have a common cigenfunction

LVf -~f

The eigenvalues E, R are known to be related by an algebraic equation

2J 4 1

R' = I1 (A-J-I

and the common eigenfunction (the 'Baker function') is an analytic function on the Riemannsurface (1) or, equivalently, a holomorphic section of a certain line bundle on (1). Until now,

the theory of finite-gap operators has drawn mostly on the analytical aspects of Riemannsurfaces and on their abstract, intrinsic geometry.

Our aim in this paper is to explain some of the extrinsic properties of the curves, line bundles,and isospectral tori (Jacobians) when those are embedded as concrete objects in a projectivespace.

There are several reasons for studying geometric realizations of the finite-gap operator theory.The classical theory ofcurves andJacobians is very beautiful, and an interesting statement aboutabstract curves and line bundles should be worth repeating about concrete representations.

Furthermore, when one integrable system, like the stationary Lax equation [L, B] = 0, is

t Permanent address: Department of Mathematics, Ohio State University, Columbus, Ohio 43210, U.S . A.

[711

,-. .. -..... ,% %

Volume I I13A, number 5 P'I IYS I(S L1111.P S 23 December 1985

THE SHAPE OF STATIONARY DISLOCATIONS

D. MI'RON and A.C. NEWELL -

Department q Mathe.l atic v. L nm er.mn tI ,IArtzna. I'it,.it. I ,52. 1 S

Recet'ed 21 October 1t)S;- accepted for publication2 ",cth r19.5 25 October 1985

It i, h,-mn that the1 ftructure o.f the tatiltonar\, kk localtOT,

%%iIuCI OcCLU[ III a!' \%I LIeC of pattern, H1(III on quIlibhiumf

s%,tem , ie en b\ a elf(-lmidar s'lmutiii of the univers~a Cros-Newell equation.

-

There has been a tremendous resurgence of interest phase 0 whose gradient is the constant wavevectorlately in the structure and properties of symmetry k. The periodicity demand gives rise to a relation

breaking instabilities in driven systems far from equi- (equivalent to the frequency dependency on ampli-librium. The planfornis and patterns which emerge tude in nonlinear oscillators)from these instabilities are observed in a wide variety o(k.AR) =0 -(2)

of physical situations, from convection in fluids, inliquid crystals and in binary mixtures to the solidifi- between the wavenumber k = Ikl. the amplitude Acation processes in undercooled liquids. Of particular and the stress parameter R which in thc case of con-interest are patterns which form in large aspect ratio vection is tihe Rayleigh number. The appearance ofsystems; for example. the Rayleigh--Bdnard convec- the modulus of k reflects the rotational symmetry.tion of a Boussinesq fluid in a box wide enough to The regions of stability of these solutions in the R. kcontain many rolls. Although for certain parameter plane have been mapped out by Busse 121 (the Busse .ranges and in infinite horizontal geometries, there balloon).exist stable, fully nonlinear, spatially periodic solu- The ('ross-- Newell theory develops a universaltions of the Oberbeck-Boussmesq equations phase equation for the slowly changing wavevector k,

a change necessitated by the influence of distant, butulx, t) =(0, A, R), ( 1) finitely distant, horizontal boundaries to which thewhich correspond to a field of inlimlely long, straight, roll axis is perpendicular (if the thermal contact be-parallel rolls, the patterns which are typically seen tween wall and fluid is good). For order one values

are much more complicated, involving patches of of R - R c R is the transition value at which rollscurved rolls, defects such as roll dislocations and first appear), the amplitude is still determined algebrai-grain boundaries. In an effort to treat the statics and cally in terms of the phase gradient by (2). In the ab-slow dynamics of these patterns, Cross and Newell sence of the mean drift effect, which we do not con-[ I developed a theory which averages over the de- sider here, the phase equation istailed local structure of the rolls and concentrates onthe global and universal properties of the pattern it- -(k)O + V(kB(k)) + F(k) (DI D 2 + D 2 D I )A

self. The starting point of the theory is the existence + ... 0 (3)and stability of the solution ( I ) representing the " -

underlying roll structure. I lere f is 27r periodic in the where the functions r(k) > 0, B(k) and Fb(k are cal-culated easily from the particular model of interest

t),partmnen ,f Applied Mathcmatic. (allech. P'daaden,. (see ref. I II) and the operators D aInd D 2 are 2k' V(\41125. t'SA + V-k and V 2, respectively. The function B(k) always

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CHAPTER 4

AN APPLICATION: THE ROLL-UP OF A FINITE VORTEX LAYER

The evolution and regularity of vorticity distributions with

varying degrees of smoothness are of great interest to those who

study the Euler equations of fluid flow. A periodically perturbed

vortex sheet, for example, is now widely believed to acquire a

curvature singularity in finite time. Using the methods developed in

Chapters 2 and 3 for the highly accurate ajoplication of boundary

integral methods to multiply connected domains, the reyularity of a

thin, periodic layer of constant vorticity is investigated

numerically. Numerical results suggest that, like the vortex sheet,

the interfaces develop a curvature singularity, but now only in

infinite time.

1. Background

The simplest model of a niyh Reynolds number shear layer is a

surface of discontinuity, or vortex sheet, between two shearing,

inviscid, irrotational fluids as pictured schematically in Figure 11.

The instability of such an interface is prototypic and is the well-

known Kelvin-Helmholtz instability. For simplicity, assume the fluid

is of uniform density, and that u + +Uj as y + ±-. Then the

linear evolution of a normal mode of amplitude A(k,t) and wave

number k is given by

68

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I IIIIII

ORDER PRESERVING APPROXIMATIONS TO SUCCESSIVE

DERIVATIVES OF PERIODIC FUNCTIONS

BY ITERATED SPLINES

M.J. Shelley

and

G. R. Baker

Department of MathematicsUniversity of ArizonaTucson, Arizona 85721

' Ph%,oca 1I) (1986) 95-112North-Holland, Amsterdam

CHAOS AND COHERENT STRUCTURES IN PARTIAL DIFFERENTIAL EQUATIONS

Alejandro ACEVES, Hatsuo ADACHIHARA. Christopher JONES, Juan (Carlos LERMAN,David W. McLAUGHLIN, Jerome V. MOLONEY* and Alan C. NEWELLApplied Mathematwcs Program. ULiu'ers5iv of Ari:ona, luson. A Z "-8021, USA

This paper addresses the possible connections hetmeen chaos, the unpredictable behavior of solutions of finite dincnonal

%'stems of ordinary differential and dillerence equations and turbulence, the unpredictable behavior of solutions of partial

differential equations. It is dedicated to Martin Kru.skal on the occasion of his 60th birthday

I. Introduction singularities in finite time like those involved in the

The chaos that occurs in p.d.e.'s collapse of Langmuir waves or in filamentation in

cannot be fathomed by legalese nonlinear optics are also candidates. For example,so we apply Occarn's razor singular solutions of the Euler equations may be

and using a laser useful in understanding the behavior of thestudy structures in ring cacities Navier-Stokes equations at high Reynolds num-

An appealing idea of modern dynamics is that hers. Singular solutions like defects and disloca-

the complicated and apparently stochastic time tions certainly do play important roles in the

behavior of large and even infinite-dimensional pattern formations arising in continuum and con-

nonlinear systems is in fact a manifestation of a densed matter physics. The key idea is that each of

deterministic flow on a low-dimensional chaotic these structures is a natural asymptotic state that,

attractor. If the system is indeed low dimensional, by virtue of the various force balances in the

it is natural to ask whether one can identify the governing equations, develops an identity which

physical characteristics such as the spatial struc- does not easily decay or disperse away.

ture of those few active modes which dominate the One can envision two types of chaos occurring.

d%,namics. Our thesis is that these modes are closely The first is a phase or weak turbulence which

related to and best described in terms of a. ymptot- arises when there is an endless competition be- -

ically robust, multiparameter solutions of the non- tween equally resilient, localized coherent struc-

linear governing equations. We find it hard to tures which are infinite time asymptotic states and

define this robust nature precisely, but loosely which are initiated at random at various parts of

speaking the idea is that these solutions are very the physical domain. Examples of this type of

stable and resilient asymptotic states. They may be turbulence are solitary waves in the one-dimen-

coherent lumps like solitons and solitary waves. sional complex envelope equation, Rayleigh-They may have the form of coherent wave packets. B36nard roll patterns with different orientations.,

They may have self-similar form. They need not and the oscillatory skew varicose states in low

necessarily be the asymptotic states which develop Prandtl number convection, it is to be expected

as tends to infinity; structures which develop that such dynamics may be low dimensional. The

*P1errinen address Dept of Physics. Iferiot-watt Uniser. second type of chaos is much more dramatic and,

,its. -dinhurgh, t.ll 4AS, UK for want of a better word, may be described as an

0167-2789/86/$03.50 Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

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