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36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 )

35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )

34 A . Khovanskit , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

(TRANS2/177)

31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Net sve taev , Editors , Mathematic s in St . Petersbur g (TRANS2/174 )

29 V . Kharlamov , A . Korchagin , G . PolotovskiY , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pes in , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )

26 S . G . Gindiki n an d E . B . Vinberg , Editors , Li e Group s an d Li e Algebras : E . B .

Dynkin's Semina r (TRANS2/169 )

25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )

24 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l Equations (TRANS2/167 )

23 Yu . Ilyashenk o an d S . Yakovenko , Editors , Concernin g th e Hilber t 16t h Proble m (TRANS2/165)

22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )

Published Earlie r a s Advance s i n Sovie t Mathematic s

21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 1994

20 R . L . Dobrushin , Editor , Probabilit y contribution s t o statistica l mechanics , 199 4

19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4

18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4

17 D m i t r y Fuchs , Editor , Unconventiona l Li e algebras , 199 3

16 Serge i Gelfan d an d S imo n Gindikin , Editors , I . M . Gelfan d seminar , Part s 1 and 2 , 1993

15 A . T . Fomenko , Editor , Minima l surfaces , 199 3 14 Yu . S . i ryashenko , Editor , Nonlinea r Stoke s phenomena , 199 2

13 V . P . Maslo v an d S . N . Samborskit , Editors , Idempoten t analysis , 199 2

12 R . Z . Khasminskit , Editor , Topic s i n nonparametri c estimation , 199 2

11 B . Ya . Levin , Editor , Entir e an d subharmoni c functions , 199 2

10 A . V . Babi n an d M . I . Vishik , Editors , Propertie s o f globa l attractor s o f partia l

differential equations , 199 2

9 A . M . Vershik , Editor , Representatio n theor y an d dynamica l systems , 199 2

8 E . B . Vinberg , Editor , Li e groups , thei r discret e subgroups , an d invarian t theory , 199 2

7 M . Sh . B irman , Editor , Estimate s an d asymptotic s fo r discret e spectr a o f integra l an d

differential equations , 199 1

6 A . T . Fomenko , Editor , Topologica l classificatio n o f integrabl e systems , 199 1

5 R . A . Minlos , Editor , Many-particl e Hamiltonians : spectr a an d scattering , 199 1

4 A . A . Suslin , Editor , Algebrai c K-theory , 199 1

3 Ya . G . Sinai , Editor , Dynamica l system s an d statistica l mechanics , 199 1

2 A . A . Kirillov , Editor , Topic s i n representatio n theory , 199 1

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American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 18 2

Advances in the Mathematical Sciences—36 (Formerly Advances in Soviet Mathematics)

Nonlinear Wave s and Wea k Turbulenc e

V. E. Zakharo v Editor

American Mathematical Societ y Providence, Rhode Island

http://dx.doi.org/10.1090/trans2/182

ADVANCES I N TH E MATHEMATICA L SCIENCE S EDITORIAL COMMITTE E

V. I . ARNOL D S. G . GINDIKI N V. P . MASLO V

Translation edite d b y A. B. Sossinsky

1991 Mathematics Subject Classification. Primar y 76B15 , 76C20; Secondary 35Q35 , 35Q53.

ABSTRACT. Th e boo k i s a collectio n o f revie w paper s o n dynamica l an d statistica l theor y o f nonlinear wav e propagatio n i n dispersiv e conservativ e media , wit h th e emphasi s o n wave s o n th e surface o f idea l flui d an d o n Rossb y wave s i n th e atmosphere . Althoug h th e boo k deal s mainl y with weakl y nonlinea r waves , i t i s fa r fro m bein g jus t a descriptio n o f standar d perturbatio n techniques. Th e ultimat e goa l o f th e boo k i s to sho w tha t th e theor y o f weakl y interactin g wave s is naturall y relate d t o suc h area s o f mathematic s a s Diophantin e equations , differentia l geometr y of waves , Poincar e norma l forms , an d th e invers e scatterin g method .

The boo k i s usefu l fo r researcher s an d graduat e student s workin g i n th e theor y o f nonlinea r waves an d it s applications .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-4113- 0

ISSN 0065-929 0

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Contents

Introduction V. E . ZAKHARO V vi i

Invariants o f Wave System s an d We b Geometr y A. M . BAL K AN D E . V . FERAPONTO V 1

Stability o f Weak-Turbulence Kolmogoro v Spectr a A. M . BAL K AN D V . E . ZAKHARO V 3 1

Energy Transfe r i n th e Spectru m o f Surfac e Gravit y Wave s b y Resonanc e Five Wave-Wav e Interaction s VALERI A . KALMYKO V 8 3

Wave Resonance s i n System s wit h Discret e Spectr a E. KARTASHOV A 9 5

Hamiltonian Formalis m fo r Rossb y Wave s L. I . PlTERBAR G 13 1

Weakly Nonlinea r Wave s o n th e Surfac e o f a n Idea l Finit e Dept h Flui d V. E . ZAKHARO V 16 7

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Introduction

The presen t boo k i s a collectio n o f paper s devote d t o differen t aspect s o f th e theory o f nonlinea r wave s propagatin g i n conservativ e media .

This theor y ha s develope d intensivel y durin g th e las t thre e decade s an d i s now a ric h an d flourishing chapte r o f mathematica l physics . Th e mos t popula r object s here ar e solitons . Th e discover y o f the Invers e Scatterin g Metho d an d th e creatio n of th e moder n mathematica l theor y o f soliton s wer e amon g th e mos t impressiv e events i n mathematic s durin g th e secon d hal f o f thi s century .

Another importan t an d broadly studied subject o f the theory of nonlinear wave s is th e formatio n o f "wav e collapses"—singularitie s arisin g i n finite tim e i n a n ini -tially smoot h medium . Bot h soliton s an d wav e collapse s ar e "strongl y nonlinear " phenomena. I n thi s boo k w e stud y anothe r clas s o f wav e phenomena—weakl y interacting waves , whic h ca n b e create d b y differen t perturbatio n methods . A t first glanc e thi s i s a traditiona l field, hardl y hoardin g somethin g intriguin g an d re -ally nontrivial . W e wil l see , however , tha t thi s i s no t th e case , an d th e theor y o f weakly interactin g nonlinea r wave s i n conservativ e medi a conceal s dee p an d diffi -cult question s an d i s connected i n an unexpecte d wa y with som e topics o f classica l mathematics.

The first question in the theory of nonlinear waves is about a universal languag e which woul d allo w u s t o stud y thes e wave s fro m a unifie d poin t o f view . A s ha s been show n durin g th e las t thirt y years , thi s languag e i s Hamiltonian formalism .

Usually, equation s describin g nonlinea r medi a ar e no t writte n a s Hamiltonia n equations. Bu t i f th e mediu m i s conservative , the y alway s hav e a hidde n Hamil -tonian structure . Findin g thi s structur e ca n b e a har d problem .

One of the basic models in meteorology, astrophysics , an d plasma physics is the so-called Obukhov-Charney-Hasegawa-Mim a equation . Thi s equatio n describe s Rossby wave s in planetary atmosphere s an d drif t wave s in plasma, an d i t ha s bee n studied analyticall y an d numericall y sinc e th e thirties . Bu t jus t recently , i n 1988 , it wa s foun d tha t thi s equatio n i s a Hamiltonia n system . Th e Hamiltonia n i s th e energy, but th e canonical variables are connected with the physical variable (strea m function) i n a very nontrivial way. Thi s interesting subject i s discussed in the articl e of L . I . Piterbar g presente d i n thi s book .

After introducin g Hamiltonian formalis m on e has to introduce normal variable s and comple x amplitude s o f propagating wave s a x, a * an d presen t th e Hamiltonia n in th e for m o f standard powe r serie s i n thes e variables .

The followin g questio n i s o f fundamenta l importance : ho w man y additiona l integrals o f motion doe s thi s syste m admit ?

vii

viii I N T R O D U C T I O N

One can distinguis h betwee n exac t integrals , whic h hol d i n al l orders o f power s of a,k, a%, an d approximat e integrals , holdin g onl y u p t o a certai n finite order . Ex -act integrals , beside s thos e o f energy an d momentum , ar e ver y rare . A s a rule , th e existence of even one "additional " exac t integra l i s a strong indicatio n tha t th e sys-tem unde r consideratio n i s integrable, lik e the well-know n Kadomtsev-Petviashvil i equation o f th e secon d typ e (KP-2) , o r "quasi-integrable" , lik e th e Kadomtsev -Petviashvili equatio n o f the first typ e (KP-1) .

On th e othe r hand , system s admittin g approximat e integral s o f motio n ar e very common . I t i s remarkabl e tha t th e existenc e o f a n approximat e integra l ca n be establishe d jus t b y analyzin g a linea r dispersio n relatio n Wk- ,- In th e crudes t classification, dispersio n relation s ar e divide d int o tw o classes : th e "deca y type " and th e "nondeca y type" . I n th e first case , the three-wav e resonan t condition s

(1) w k = w kl + w k2, k = ki + k 2,

describe a rea l manifold . In the second case , equations (1 ) have no solution. I n this case the Hamiltonia n

equations alway s hav e a n additiona l approximat e integral : th e tota l "wav e action "

/ \dk\ 2dk

and the initia l variables a k ar e not th e mos t convenien t an d adequat e fo r describin g nonlinear wav e effects . T o find bette r variables , on e ha s t o perfor m a canonica l transformation eliminatin g th e cubi c term s i n th e Hamiltonian .

In thes e ne w variable s th e interactio n Hamiltonia n (th e "effective " Hamilton -ian) resembles the first term of the "Poincar e normal form" fo r a system of nonlinear oscillators. Bu t th e Poincar e norma l for m exist s onl y i n system s wit h a discret e number o f degree s o f freedo m and , strictl y speaking , doe s no t exis t fo r a syste m of wave s i n infinit e space . Th e structur e o f th e effectiv e Hamiltonia n i s crucia l for understandin g th e lowest-orde r nonlinea r effects . I t coul d b e quit e differen t i n different physica l systems .

In m y articl e "Weakl y nonlinea r wave s o n th e surfac e o f a n idea l finite dept h fluid" thi s structur e i s studied fo r a problem importan t i n applications , namely , th e gravity wave s o n th e surfac e o f a n idea l fluid o f finite dept h (especiall y o n shallo w water). Th e cas e o f infinitel y dee p wate r wa s studie d earlie r i n 1968 . Th e reade r will se e ho w differen t thes e tw o case s are . Gravit y wave s o n shallo w wate r ar e described i n th e first approximatio n b y th e KP- 2 equation , whic h i s a completel y integrable system . Thi s remarkabl e fac t lead s t o cancellatio n o f th e leadin g term s in th e effectiv e Hamiltonia n an d a n essentia l chang e o f th e entir e physica l pictur e of wave interaction .

Dispersion relation s o f th e deca y typ e admi t furthe r classification . On e ca n distinguish betwee n degenerat e an d nondegenerat e dispersio n laws . I n th e first case, on e can find a function fk whic h i s not a linea r combinatio n o f Wk and & , but satisfies th e equatio n

fk ~ fk x + fk 2

on th e resonan t manifol d (1) . I n th e secon d case , the dispersio n relatio n i s nonde -generate.

INTRODUCTION i x

If some a\z i s degenerate, on e can construc t a n additiona l approximat e integra l of the for m

I=j fk\a k\2dk + -'- .

The numbe r o f suc h integral s coincide s wit h th e numbe r o f linearl y independen t functions fk-

It i s quite nontrivia l tha t som e dispersion relation s describin g wav e interactio n from a physica l poin t o f view ar e degenerate .

For instance , Rossb y wave s hav e a degenerat e dispersio n relatio n an d admi t one additiona l approximat e integral . Thi s fac t play s a ver y importan t rol e i n th e dynamics o f weakl y nonlinea r Rossb y waves . I t i s obviou s tha t th e proble m o f classifying degenerat e dispersion relation s i s one of the mos t fundamenta l problem s in the theory of nonlinear wave interaction. Th e solution of this problem was greatly advanced i n the articl e b y A. Balk an d E . Ferapontov , "Invariant s o f wave system s and we b geometry" . I t happene d tha t thi s proble m ca n b e formulate d i n term s o f "web geometry" , a chapte r o f Classica l Differentia l Geometr y activel y develope d by Blaschk e an d hi s schoo l i n th e thirties .

Through th e us e o f web geometry , Bal k an d Feraponto v foun d ne w interestin g classes of degenerate dispersio n relations , no t onl y for three-wave , bu t als o for four -wave interactions .

Weakly nonlinea r dynamica l effect s ar e als o studie d i n th e articl e b y E . Kar -tashova "Wav e resonance s i n system s wit h discret e spectra" . Thi s articl e i s ver y interesting fo r th e followin g reasons . Traditionally , nonlinea r wave s ar e studie d i n infinite space , wher e th e wav e vecto r & i s a continuou s variable . Bu t i n practice , all system s ar e finit e an d th e eigenfrequenc y spectru m i s alway s discrete . I n th e simplest cas e o f a rectangula r domain , wav e number s for m a regula r lattice , an d the resonan t condition s (1 ) o r thei r generalizatio n becom e a system o f Diophantin e equations. I n som e case s (fo r instance , fo r capillar y waves ) the y hav e n o solution s at all . Bu t i n othe r importan t case s (Rossb y waves , gravity waves ) thes e equation s have nontrivia l solution s whic h ca n b e foun d b y th e method s o f numbe r theory . This importan t wor k wa s don e b y Kartashov a an d th e result s ar e collecte d i n thi s article.

So far , w e hav e discusse d th e dynamica l aspect s o f weakl y nonlinea r wav e in -teraction. Bu t i n man y case s th e numbe r o f excite d degree s o f freedo m i s s o larg e that th e wave s hav e t o b e describe d statistically . Th e statistica l theor y o f weakl y nonlinear dispersiv e wave s i s the theor y o f weak turbulence .

The mai n tool s here ar e kinetic equations fo r th e average d square s o f wave am -plitudes, similar to the Boltzmann equatio n in kinetic gas theory. Kineti c equation s describing wea k wav e turbulenc e hav e exac t power-lik e solutions : wea k turbulen t Kolmogorov solutions . Th e theor y o f these equation s i s now wel l developed. Thes e solutions, althoug h stationary , describ e spectr a ver y fa r fro m thermodynamica l equilibrium, an d th e questio n abou t thei r stabilit y i s quit e nontrivial . Th e prob -lem o f th e stabilit y o f weak-turbulenc e Kolmogoro v spectr a i s th e subjec t o f th e article b y Bal k an d Zakharov . W e sho w tha t thi s proble m ca n b e reduce d t o th e calculation o f the rotatio n indice s fo r a certain functio n analyti c i n an infinit e stri p parallel to the imaginary axis . Fo r al l concrete cases this problem ca n be efficientl y solved. Numerica l calculation s o f certain integral s ca n b e don e i f necessary .

The articl e b y Kalmyko v i s devote d t o th e numerica l solutio n o f th e "five -wave" kineti c equatio n fo r gravit y wave s o n dee p water . I t i s interestin g fro m

x INTRODUCTIO N

the theoretica l poin t o f vie w becaus e five-wave interactio n i s th e proces s o f th e lowest orde r tha t violate s conservatio n o f th e approximat e integral , namel y th e wave action . Th e stud y o f thi s proces s i s the questio n o f paramoun t interest .

The articles collected in the book, being rather diverse , are connected by a com-mon spirit . Al l o f them wer e reporte d i n m y semina r o n nonlinea r wav e problems , which wa s hel d i n Mosco w fo r mor e tha n te n year s (1984-1994) .

V. E . Zakharo v

Tuscon, Arizona , 199 7