Instability of large solitary water waves

Transcription

1 Istability of larg solitary watr wavs Zhiwu Li Mathmatics Dpartmt Uivrsity of Missouri Columbia, MO 65 USA Abstract W cosir th liariz istability of D irrotatioal solitary watr wavs. Th maxima of rgy a th travl sp of solitary wavs ar ot obtai at th highst wav, which has th 0 gr agl at th crst. Ur th assumptio of o-xistc of scoary bifurcatio which is co rm umrically, w prov liar istability of solitary wavs which ar highr tha th wav of maximal rgy a lowr tha th wav of maximal travl sp. It is also show that thr xist ustabl solitary wavs approachig th highst wav. Th ustabl wavs ar of larg amplitu a thrfor this typ of istability is ot captur by th approximat mols riv ur small amplitu assumptios. For th proof, w itrouc a family of olocal isprsio oprators, which ar rlat to th bifurcatio of solitary wavs. A cotiuity argumt with a movig krl formula is us to stuy ths isprsio opartors to yil th istability critria. Itrouctio Prlimiaris. Th watr-wav problm i its simplst form cocrs twoimsioal motio of a icomprssibl ivisci liqui with a fr surfac, act o oly by gravity. Suppos, for itss, that i th (x; y)-cartsia cooriats gravity acts i th gativ y-irctio a that th liqui at tim t occupis th rgio bou from abov by th fr surfac y (t; x) a from blow by th at bottom y h, whr h > 0 is th watr pth. I th ui rgio f(x; y) : h < y < (t; x)g, th vlocity l (u(t; x; y); v(t; x; y)) satis s th icomprssibility x u y v 0 (.) a th Eulr quatio t u + u@ x u + v@ y x t v + u@ x v + v@ y y P g; (.)

2 whr P (t; x; y) is th prssur a g > 0 ots th gravitatioal costat of acclratio. Th kimatic a yamic bouary coitios at th fr surfac fy (t; x)g t + u@ x a P P atm (.3) xprss, rspctivly, that th bouary movs with th vlocity of th ui particls at th bouary a that th prssur at th surfac quals th costat atmosphric prssur P atm. Th imprmability coitio at th at bottom stats that v 0 at fy hg: (.4) I this papr w cosir th irrotatioal cas with curl v 0, for which th Eulr quatio (.) (.4) is ruc to th + jrj + g c (t) whr is th vctor pottial such that (u; v) r. Th local wll-posss of th full watr wav problm was prov by Wu ([67]) for p watr a by Las for watr of it pth ([34]). W cosir a travlig solitary wav solutio of (.) (.4), that is, a solutio for which th vlocity l, th wav pro l a th prssur hav spactim pc (x + ct; y), whr c > 0 is th sp of wav propagatio. With rspct to a fram of rfrc movig with th sp c, th wav pro l appars to b statioary a th ow is stay. It is traitioal i th travligwav problm to th rlativ stram fuctio (x; y) a vctor pottial (x; y) such that: x v; y u + c (.5) a Th bouary coitios at i ity ar x u + c; y v: (.6) (u; v)! (0; 0) ; (x)! 0, as jxj! +: Th solitary wav problm for (.) (.4) is th ruc to a lliptic problm with th fr bouary fy (x)g ([4]): Fi (x) a (x; y), i f(x; y) : < x < +; h < y < (x)g, such that 0 i h < y < (x); (.7a) 0 o y (x); (.7b) jr j +gy c o y (x); (.7c) ch o y h; (.7) with r! (0; c), (x)! 0, as jxj! +:

3 First w giv a summary of th xistc thory of solitary watr wavs. Lavrtiv ([35]) got th rst proof of th xistc of small solitary wavs by stuyig th log wav limit. A irct costructio of small solitary wavs was giv by Fririchs a Hyrs ([6]), a thir proof was rarss by Bal ([8]) via th Nash-Mosr mtho. Th xistc of larg amplitu solitary wavs was show by Amick a Tola ([4]). Th highst wav was also show to xist by Tola ([65]), a its agl at crst was show to b 0 gr (Stoks s Cojctur) by Amick, Tola a Frakl ([6]). Th symmtry of solitary wavs was stui by Craig a Strbrg ([3]). Plotikov ([58]) stui th scoary bifurcatio a show that solitary wavs ar ot uiqu for crtai travllig sp. Th particl trajctory for solitary wavs was stui by Costati a Eschr ([]). W list som proprtis of th solitary wavs which will b us i th stuy of thir stability. Dot th Frou umbr by a th Nkrasov paramtr by F c p gh ; 6ghc qc 3 ; whr q c is th (rlativ) sp of th ow at th crst. W ot that is th bifurcatio paramtr us i [4]. Th highst wav corrspos to + sic q c 0:Th followig proprtis of solitary wavs ar prov: (P) ([4]) Thr xists a curv of solitary wavs that ar symmtric, positiv ( > 0) a mootoically cay o ithr si of th crst, with th paramtr 6 ; +. Wh % +, th solitary wavs t to th highst wav with th 0 gr agl at th crst. Wh & 6 ; th solitary wavs t to th small wavs costruct i [6] a [8]. Morovr, w hav r! (0; c), (x)! 0 xpotially as jxj! +: Blow, w call this solitary wav curv th primary brach. (P) ([4], [53]) Ay positiv a symmtric solitary wav which cays mootoically o ithr si of its crst is suprcritical, that is, F > (c > p gh). Th small wav limit corrspos to F &. (P3) ([3]) Ay suprcritical solitary wav (F > ) is symmtric, positiv a cays mootoically o ithr si of its crst. Morovr, ay otrivial solitary wav curv coct to th primary brach must hav F >. (P4) ([58]) For small amplitus wavs with 6, thr is o scoary bifurcatio o th primary brach. Wh th highst wav is approach, that is, wh! +, thr ar i itly may poits o th primary brach which ar ithr a scoary bifurcatio poit or a turig poit whr c 0 () 0: Th proprty (P4) is sstially what was prov i [58], though our statmt abov aapts th xplaatio i [5, p. 45]. Morovr, umrical vics ([7], [33]) iicat that th followig assumptio hols tru: (H) Thr ar o scoary bifurcatio poits o th primary brach. Ur th assumptio (H), proprty (P4) implis that thr ar i itly may turig poits whr c 0 () 0. So th travl sp c os ot always 3

4 icras with th wav amplitu, a this i rs gratly from KDV a othr approximat mols for which th highr wavs travl fastr. Mor prcisly, for full solitary watr wavs th travl sp obtais its maximum bfor th highst wav a th it bcoms highly oscillatory ar th highst wav. This fact was rst obsrv from umrical computatios ([7], [48]), th co rm by th asymptotic aalysis ([49], [5]). I, almost all physical quatitis (i.. rgy a momtum) o ot achiv thir maxima at th highst wav, a ar highly oscillatory arou th highst wav (s abov rfrcs). This fact turs out to imply th istability of larg solitary wavs, which was rst iscovr from umrical computatios ([63]) a is rigorously prov i this papr. Mai rsults. Dot by th rst turig poit whr c () obtais its maximum a, by ~ th rst a also th absolut maximum poit of E (), whr Z Z (x) E () u + v Z yx + R g x: (.8) R h is th rgy of th solitary wav with th paramtr. Numrical computatios ([48], [63], [5]) iicat that > ~, a ~ is th oly critical poit of E () i 6 ;. W stat it as aothr hypothsis: (H) Th rgy maximum is achiv o th primary brach bfor th wav of th maximal travl sp (th rst turig poit). Thorm Ur th assumptios (H) a (H), th solitary wav is liarly ustabl wh (~ ; ), whr a ~ ar th maxima poits of th travl sp a rgy, rspctivly. Th liarly istability is i th ss that thr xists a growig mo solutio t [ (x) ; (x; y)] ( > 0) to th liariz problm (.), whr (x) ; (x; y) C \ H k for ay k > 0: Our xt thorm shows that thr xist ustabl solitary wavs approachig th highst wav. Thorm Ur th assumptio (H), thr xists i itly may itrvals I i ( ; +); (i ; ; ) with lim! max fj I g +, such that solitary wavs with th paramtr I i ar liarly ustabl i th ss of Thorm. Thorm suggsts that th highst wav ( ) costruct i [6] is ustabl. This cotrasts with th stability of pak solitary wavs i som shallow watr wav mols ([], [46]). Numrical vics ([63], [5]) suggst that solitary wavs ar spctrally stabl wh 6 ; ~, a liarly ustabl wh > ~, at last bfor th rst fw turig poits whr th computatios ar rliabl. W ot that th amplitu of th maximal rgy wav with th paramtr ~ is alray clos to th maximal hight ([63]). So th ustabl wavs prov i Thorms a ar of larg amplitu, a thrfor this typ of istability os ot appar i approximat mols that ar riv ur th small amplitu assumptios. Numrical vics ([64]) also suggst that 4

5 this larg amplitu istability ca la to wav brakig. Such wav brakig iuc by istability is also us to xplai th brakig wavs approachig bachs ([5], [56], [57]). Mor iscussios of ths issus ar fou i Rmarks a (Sctio 5). Th proof of Thorms a also has som implicatios for th spctral stability of solitary wavs with < ~. W ot that th solitary wavs of full watr wavs ar show ([], [3]) to b highly i it rgy sals ur th costraits of costat momtum, mass tc. Thrfor, thir stability caot b stui by th traitioal mtho of provig (costrait) rgy miimizrs as i may mol quatios such as th KDV quatio ([9], [3]). So far thr ar fw ctiv mthos for provig oliar stability of rgy sals. So aturally, th rst stp is to stuy thir spctral stability, amly, to show that thr os ot xist a xpotially growig solutio to th liariz problm. Th followig thorm is usful for this purpos. Thorm 3 Assum th hypothsis (H). Suppos that thr is a squc of purly growig mos t [ (x) ; (x; y)] ( > 0) to th liariz problm for solitary wavs with paramtr f g, a! 0+,! 0 whr 0 is ot a turig poit, th w ( 0) 0. By th abov thorm, if oscillatory istability ca b xclu, that is, ay growig mo is show to b purly growig, th th trasitio of istability ca oly happ at th rgy xtrma or turig poits. Numrical rsults i [63], [5] justify that th growig mos fou ar i purly growig for solitary wavs bfor th rst fw turig poits. If aitioally th stability of small solitary wavs is prov, th it follows that th solitary wavs ar stabl up to th wav of maximal rgy. Commts a ias of th proof. First, w commts o rlat rsults i th litratur. I [59], Sa ma cosir th spctral stability of prioic wavs i p watr (Stoks wavs), ur prturbatios of th sam prio (suprharmoic prturbatios). Th pictur of suprharmoic istability of Stoks wavs ([6], [6]) is similar to that of th istability of solitary wavs. Th approach of [59] is to tak th it mo trucatio of th liariz Hamiltoia formulatio of Zakharov ([68]) a stuy th igvalu problm for th matrix obtai. By assumig th xistc of a squc of purly growig mos with th growth rat! 0+ a paramtrs! 0, th solvability coitios ar chck to th sco orr to show that 0 must b a rgy xtrmum. That is, a aalogu of Thorm 3 was stablish i [59] for Stoks wavs. Howvr, th aalysis i [59] is at a rathr formal lvl. First, Zakharov s Hamiltoia formulatio has a highly i it quaratic form that is ubou from both blow a abov. This is u to th i itss of th rgy fuctioal of th pur gravity watr wav problm ([]) as mtio bfor. So it is uclar how to pass th it trucatio rsults i [59] to th origial watr wav problm. Scoly, a implicit assumptio i [59] is that th trucat matrix has th zro igvalu of gomtric multiplicity. It is uclar how to chck a rlat this assumptio to th proprtis of stay wavs. For 5

6 solitary wavs, th trucatio approach of [59] sms i cult to apply bcaus of th ubou omai. Rctly, i [3], [33], Kataoka rcovr Sa ma s formal rsult (or aalogus of Thorm 3) for prioic watrs i watr of it pth a for itrfacial solitary wavs i a i rt way. Th aalysis of [3], [33] is agai formal a of similar atur as [59]. That is, by assumig th xistc of purly growig mos with vaishig growth rats, th rst two solvability coitios wr chck to show that th limitig paramtr is a rgy xtrmum. W ot that i th abov paprs of Kataoka a Sa ma, th xistc of purly growig mos with vaishig growth rats was oly assum, but ot prov, a thir claims about th xchag of stability at rgy xtrma hig o th uprov assumptio that all growig mos ar purly growig. I this papr, w provi th rst proof of th xistc of ustabl solitary wavs. To obtai th complt pictur of stability of solitary wavs i th whol brach, o still to xclu th oscillatory istability a to prov th stability of small wavs, for which th w formulatio i this papr might also b usful. Blow, w bri y iscuss mai ias i th proof of Thorms a. To avoi th issu of i itss mtio abov, w o ot aapt Zakharov s Hamiltoia formulatio i trms of th vctor pottial o th fr surfac a th wav pro l. W us th liariz systm riv i [3], i trms of th i itsimal prturbatios of th wav pro l a th stram fuctio rstrict o th stay surfac S :Th w furthr ruc this systm to gt a family of oprator quatios A ( j S ) 0, whr > 0 is th ustabl igvalu to b fou:th oprator A is th sum of th Dirichlt-Numa oprator a a bou but olocal oprator. Th ia of abov ructio is to rlat th igvalu problms to th lliptic typ problms for stay wavs. Th hoograph trasformatio is th us to gt quivalt oprators A - o th whol li. Th xistc of a purly growig mo is quivalt to som > 0 such that th oprator A has a otrivial krl. This is achiv by usig a cotiuity argumt to xploit th i rc of th spctra of A ar i ity a zro. Th zro-limitig oprator A 0 is particularly itrstig that th stuy of bifurcatio of stay wavs ([58]) ca b ruc to stuy xactly th sam oprator (s Lmma 4.). So our stuy also stablishs som rlatio btw th bifurcatio a stability of solitary wavs. Th ia of itroucig olocal isprsio oprators A with a cotiuity argumt to gt istability critria origiats from our prvious works ([40], [39], [38]) o D ial ui a D lctrostatic plasma, which hav also b xt for galaxy yamics [8] a 3D lctromagtic plasmas [4], [4]. O w issu i th currt cas is th i uc of th symmtry of th problm. Mor spci cally, w to ursta th movmt of th krl of A 0 that is u to th traslatio symmtry, ur th prturbatio of A 0 to A for small. This is obtai i a movig krl formula (Lmma 5.) which also implis Thorm 3. Th covrgc of A to A 0 is rathr wak, so th usual prturbatio thory os ot apply a th asymptotic prturbatio thory by Vock a Huzikr ([66]) is us to stuy prturbatios of th igvalus of A 0. A importat tchical part i our proof is to us th suprcritical prop- 6

7 rty F > a th cay of solitary wavs to obtai a priori stimats a gai crtai compactss. Th tchiqus vlop i this papr hav b rctly xt to show istability of larg Stoks wavs ([43]) a gt istability critria for prioic a solitary wavs of vry gral isprsiv wav quatios ([44], [45]). This papr is orgaiz as follows. I Sctio, w giv th formulatio of th liariz problm a riv th olocal isprsio oprators A. Sctio 3 is vot to stuy proprtis of th oprators A, i particular, thir sstial spctrum. I Sctio 4, w apply th asymptotic prturbatio thory to stuy th igvalus of A for ar 0. I Sctio 5, w riv a movig krl formula a prov th mai thorms. Som importat formula ar prov i Appix. Formulatio for liar istability I this Sctio, a solitary wav solutio of (.7) is hl x, as such it srvs as th uisturb stat about which th systm (.) (.4) is liariz. Th rivatio is prform i th movig fram of rfrcs, i which th wav pro l appars to b statioary a th ow is stay. Lt us ot th uisturb wav pro l a rlativ stram fuctio by (x) a (x; y), rspctivly, which satisfy th systm (.7). Th stay rlativ vlocity l is (u (x; y) + c; v (x; y)) ( (x; y); x (x; y)); a th stay prssur P (x; y) is trmi through Lt a P (x; y) c jr (x; y)j gy: (.) D f(x; y) : < x < + ; h < y < (x)g S f(x; (x)) : < x < +g ot, rspctivly, th uisturb ui omai a th stay wav pro l. Lt us ot ((t; x); u(t; x; y); v(t; x; y); P (t; x; y)) to b th i itsimal prturbatios of th wav pro l, th vlocity l a th prssur rspctivly. Th stram fuctio prturbatio is (t; x; y), such that (u; v) ( y ; x ) : Th liariz watr-wav problm was riv i [3], a it taks th followig form i th irrotatioal cas: 0 i D ; t ( ) 0 o S ; (.b) 7

8 whr P + P 0 o S ; ) P 0 o S ; x 0 o fy hg; x f + y f y f x f ot th tagtial a ormal rivativs of a fuctio f (x; y) o th curv fy (x)g. f(x) x f(x; (x)). Not that th abov liariz systm may b viw as o for (t; x; y) a (t; x). I, P (t; x; (x)) is trmi through (.c) i trms of (t; x) a othr physical quatitis ar similarly trmi i trms of (t; x; y) a (t; x). A growig mo rfrs to a solutio to th liariz watr-wav problm (.a)-(.) of th form ((t; x); (t; x; y)) ( t (x); t (x; y)) a P (t; x; (x)) t P (x; (x)) with R > 0. For a growig mo, th liariz systm (.) bcoms a th followig bouary coitios o S ; 0 i D (.3) (x) + x ( (x; (x))(x)) x (x; (x)); (.4) P (x; (x)) + P (x; (x))(x) 0; (.5) (x) + x ( (x; (x)) (x)) x P (x; (x)): (.6) W impos th followig bouary coitio o th at bottom (x; h) 0; (.7) from which (.) follows. I summary, th growig-mo problm for a solitary watr-wav is to a otrivial solutio of (.3)-(.7) with R > 0. Blow, w look for purly growig mos with > 0 a ruc th systm (.3)-(.7) to o sigl quatio for j S. For simplicity, hr a i th squl w itify (x) with (x; (x)) a (x) with (x; (x)), tc. First, w itrouc th followig oprator C + x ( (x)) x : (.8) Not that > 0 i D by th maximum pricipl a Hopf s pricipl ([3]), a th fact that u + c! c as jxj!. Thus c 0 (x; (x)) u + c c ; (.9) 8

9 for som costat c 0 ; c > 0. D ig th oprator D x ( (x)(x)) ; w ca writ C as C D (x) (x) : (.0) Dot L (S ) to b th wight L spac o S. Bcaus of th bou (.9), L (S ) a L (S ) ar orm quivalt. Not that th oprator D is ati-symmtric o L (S ). Lmma. For > 0; th oprator E ; : L (S )! L (S ) by E ; D : Th, (a) Th oprator E ; is cotiuous i, E ; L ; (.) (S)!L (S) a E ; L : (.) (S)!L (S) (b) Wh! 0+, E ; covrgs to 0 strogly i L (S ). (c) Wh! +; E ; covrgs to strogly i L (S ). Proof. Dot M ; R to b th spctral masur of th slf-ajoit oprator R id o L (S ). Th E ; Z L R Z i km k L km k L kk L R a (.) follows. Similarly, w gt th stimat (.). To prov (b), w tak ay L (S ) a ot th fuctio () to b such that () 0 for 6 0 a (0). Th by th omiat covrgc thorm, wh! 0+; E ; Z L R i km k L Z! ()km k L M f0g : L R Not that M f0g is th projctor of L to kr D f0g. So M f0g 0 a E ;! 0 i L. Th proof of (c) is similar to that of (b) a w skip it. By th abov lmma a th bou (.9) o ; w hav 9

10 Lmma. For > 0, th oprator C : L (S )! L (S ) by (.8) has th followig proprtis: (a) C L(S)!L (S) C; for som costat C ipt of : (b) Wh! 0+, C covrgs to (x) strogly i L (S ). (c) Wh! +; C covrgs to 0 strogly i L (S ). By usig th oprator C, th growig mo systm (.3)-(.7) is ruc to (x) + C P (x) C 0; o S ; (.3) 0 i D ; (x; h) 0: W th followig Dirichlt-Numa oprator N : H (S )! L (S ) by N f (@ y f x f ) (x; (x)) ; whr f is th uiqu solutio of th followig Dirichlt problm for f H (S ) f 0 i D ; f j S f; f (x; h) 0: Th th xistc of a purly growig mo is ruc to som > 0 such that th oprator A by A N + C P (x) C : H (S )! L (S ) (.4) has a otrivial krl. Not that if w ot by f th holomorphic cojugat of f i D, f x f. This motivats us to a aalogu of th Hilbrt trasformatio as i [58], by (C f) (x) Z x 0 N f x: Th th oprator N ca b writt as N x C. From th itio, C f + if a f ic f ar th bouary valus o S of som aalytic fuctios i D. Blow, w furthr ruc th oprator A to o o th ral li. First, w th holomorphic mappig F : D! R ( ch; 0) by F (x; y) ( (x; y) ; (x; y)). W ot (; &) c ( ; ) D 0 R ( h; 0) a th mappig G : D 0! D by G (; &) (x (; &) ; y (; &)) F (c; c&) : 0

11 Th at Dirichlt-Numa oprator N : H (R)! L (R) is by N & f j f&0g ; whr f is th solutio of th followig Dirichlt problm for f H (R) f 0 i D 0 ; f j f&0g f; f j f& hg 0: Similarly, w th oprator C by Cf R x 0 N f x. Th N C a Cf + if or f icf ar th bouary valus o f& 0g of aalytic fuctios i D 0. Morovr, N is a Fourir multiplir oprator with th symbol ([3]) (k) To sparat th uiform ow (c; 0) ; w rwrit Dot k tah (kh) : (.5) (x (; &) ; y (; &)) (; &) + (x (; &) ; y (; &)) : w () y (; 0) : (.6) Th w ca st x (; 0) Cw by aig a propr costat to th vctor pottial. Th mappig G rstrict o f& 0g iucs a mappig B : R!S by B () ( + Cw; w). Dot z x + iy a p + i&, th u + c iv ( + i ) z So o f& 0g, w gt whr c p z c z p c x + i@ y : u + c c + N w jw j ; v c w0 jw j (.7) W ( x + i@ y ) j f&0g + Cw 0 + iw 0 + N w + iw 0 (.8) a 0 ots th rivativ. From (.7), + N w (u + c) c (u + c) ; + v a thus by (.9), thr xists c ; c 3 > 0 such that c < + N w < c 3 : (.9)

12 W th oprator B : L (S )! L (R) by (Bf) () f (B ()) f ( + Cw () ; w ()), for ay f L (S ): Sic BC CB a B ( + N w) B x ; w hav BN B B x C B + N w C ( + N w) N ; a Hr, BA B BN B + BC B BP B BC B ~C BC B + N w N + ~ C P () ~ C : + + N w ( ()) + N w a w us (); P () to ot (B ()) ; P (B ()) tc. For > 0; w th oprator A : H (R)! L (R) by A N + ( + N w) ~ C P () ~ C : Th th xistc of a purly growig mo is quivalt to som > 0 such that th oprator A has a otrivial krl. 3 Proprtis of th oprator A I this sctio, w stuy th spctral proprtis of th oprator A. First, w hav th followig stimat for th Dirichlt-Numa oprator N : Lmma 3. Thr xists C 0 > 0, such that for ay (0; ) a f H (R) ; w hav (N f; f) ( ) h kfk L + C 0 kfk H ; whr kfk H Z ( + jkj) ^f (k) k a ^f (k) is th Fourir trasformatio of f. Proof. By th itio (.5), Z k (N f; f) tah (kh) ^f Z (k) k jkj tah (jkj h) ^f (k) k:

13 It is asy to chck that th fuctio x h (x) tah (xh) ; x 0 satis s h (x) h 0 a lim h (x) h x! x So thr xists K > 0, such that h (x) (N f; f) h ( ( Z ^f (k) k + Z jkjk ) h kfk L + h kfk L + h ) h kfk L + mi h ; Kh ; : h x, wh x > K. Thus jkj ^f (k) k Z jkjk Z This provs th Lmma with C 0 mi h ; Kh ;. W hav th followig proprtis for th oprator C ~. ^f (k) k + Z ( + jkj) ^f (k) k jkjk Lmma 3. For > 0, th oprator ~ C : L (R)! L (R) by (.8) satis s: (a) ~ C L (R)!L (R) C; for som costat C ipt of : (b) Wh! 0+, ~ C covrgs to () strogly i L (R). (c) Wh! +; ~ C covrgs to 0 strogly i L (R). Proof. By (.9), th oprator B a B ar bou. Sic ~ C BC B, th abov lmma follows irctly from Lmma.. To simply otatios, w ot b () + N w a th oprators ~D b () ( ()) a E ~ ; (x) D ~ : wight spac L b (R). Sim- Th oprator D ~ is ati-symmtric i th b ilar to th proof of Lmma., w hav jkj ^f (k) k Lmma 3.3 (a) For ay > 0; E ~ ; L ; (3.) b!l b a E ~ ; : (3.) L b!l b (b) Wh! 0+, ~ E ; covrgs to 0 strogly i L b. (c) Wh! +; ~ E ; covrgs to strogly i L b. 3

14 Th oprator C ~ ca b writt as ~C ~ () E ~ ;+ () : Propositio For ay > 0, w hav ss A z j R h g c : (3.3) W ot that 0 : g h c > 0 (3.4) by Proprty (P), so th abov Propositio shows that th sstial spctrum of A lis o th right half pla a is away from th imagiary axis. To prov Propositio, w th followig lmmas. Lmma 3.4 For ay u H (R), w hav (i) For ay > 0, ~ E ; u H C kuk H ; (3.5) for som costat C ipt of. Blow, lt F () b a x bou fuctio that cays at i ity. Th (ii) Giv > 0; for ay " > 0, thr xists a costat C " such that F E ~ ; u L u " kuk H + C " + (iii) For ay " > 0, thr xists " > 0, such that wh 0 < < ", F E ~ ; u " kuk L H. L (iv) For ay " > 0, thr xists " > 0, such that wh > " ; F E ~ ; u " kuk L H. o Proof. Proof of (i): Dot ~M ; R to b th spctral masur of th slf-ajoit oprator ~ R with th orm ~H s i D ~ o L b. For s 0, w th spac u L b j ~ s o R u L b kuk ~H s kuk L + b ~ R s Z u kuk L L + b b : jj s k M ~ uk L b R ; 4

15 whr R ~ s is th positiv slf-ajoit oprator by R jj s M ~. W claim that th orm kk ~H s is quivalt to th orm kk H s, for 0 s. Wh s 0, H ~ 0 L b a H0 L. Sic b a ar bou with positiv lowr bous, kk L a k k b L ar quivalt. Wh s, w hav kuk ~H kuk L b + Z b ( u) b x! ; which is clarly quivalt to kuk H, agai u to th bous of b a. Wh 0 < s <, th spacs H ~ s (H s ) ar th itrpolatio spacs of H ~ 0 (H 0 ) a H ~ (H ): So by th gral itrpolatio thory ([]), w gt th quivalc of th orms kk ~H s a kk H s. Thus, thr xists C ; C > 0, such that C kuk ~H kuk H C kuk ~H : (3.6) Sic R ~ a E ~ ; ar commutabl, w hav E ~ ; u ~H ~ E ; u + L E ~ ; R ~ u b kuk L + b R ~ L b u kuk ~H : L b Th stimat (3.5) follows from abov a (3.6). Proof of (ii): Suppos othrwis, th thr xists " 0 > 0 a a squc fu g H (R) such that F E ~ ; L u " 0 ku k H + u + : L W ormaliz u by sttig F E ~ ; L u. Th ku k H ; u " 0 + L : So thr xists u H, such that u! u wakly i H u. Sic +! 0 strogly i L, w hav u 0. Thus v E ~ ; u covrgs to 0 wakly i L. By (i), kv k H C ku k H C " 0 : Lt R C 0 b a cut-o fuctio for fjj Rg. W writ F F R + F ( R ) F + F : 5

16 Th kf v k L C max jf ()j ku k L C max jjr " jf ()j 0 jjr ; wh R is chos to b big ough. Sic F has a compact support a H,! L is locally compact, so F v! 0 strogly i L. Thus, wh is larg ough, kf v k L kf v k L + kf v k L 3 4. This is a cotraictio to th fact that kf v k L F E ~ ; L u. Proof of (iii): Suppos othrwis, th thr xists " 0 > 0 a a squc fu g H (R) ;! 0+; such that F E ~ ; L u " 0 ku k H : Normaliz u byf E ~ ; L u. Th ku k H " 0. Lt u! u wakly i H. Th for ay v L, w hav ~E ; u ; v u ; b E ~ ; bcaus by Lmma 3.3, b E ~ ; v b v b! 0;! 0 strogly i L wh! 0+. So ~ E ; u! 0 wakly i L, a this las to a cotraictio as i th proof of (ii). Proof of (iv) is th sam as that of (iii), xcpt that w us th strog covrgc ~ E ;! 0 wh!. Lmma 3.5 Cosir ay squc fu g H (R) ; ku k ; supp u fj jj g : Th for ay complx umbr z with R z < 0, w hav R A z u ; u 4 0; wh is larg ough. Hr, 0 is by (3.4). Proof. W hav R A z u ; u (N u ; u ) R z + R For 0 < < (to b x latr), by Lmma 3. b ~ C P () ~ C u ; u : (3.7) (N u ; u ) ( ) h + C 0 ku k H : (3.8) 6

17 Not that by (.) P j S g y jr j g ( x xy + y ) g + ( xx + x xy ) g + x ( x) : Dot P () g + () ~a () ; ~a () x ( x) () : (3.9) Th ~a () cays xpotially wh jj!. W hav b C ~ P () C ~ u ; u Dot g b E ~ ;+ ~ E ;+ u ; u + b E ~ ;+ ~a E ~ ;+ u ; u T + T : ~ b () b ; ~c () c : (3.0) Th ~ b; ~c ts to zro xpotially wh jj!. Th rst trm ca b writt as T g b E ~ ;+ u ; E ~ ; u g b E ~ ;+ u ; E ~ ; u g b ; E ~ ;+ u ; E ~ ; u T + T ; whr i th abov w us th fact that th oprator D ~ is ati-symmtric i th spac L b. I th rst of this papr, w us C to ot a gric costat i th stimats. By Lmma 3. a th assumptio that supp u fj jj g, 7

18 w hav T g E ~ ;+ g Z g u L b u L b u L b E ~ ; b Z ju j b ju j 3 L u b g c 3 + C max ~ b () ~c () + ~ b () + j~c ()j jj c + C max jj. g c + O Sic ; E ~ ;+ ~ b () ~c () + ~ b () + j~c ()j ku k L h ~c; ~ E ;+i ~c ~ E ;+ ~ E ;+ ~c; w hav T g b ; E ~ ;+ L u E ~ ; L u C ~c E ~ L ;+ u + E ~ ;+ ~c L u : Sic ~c () cays at i ity, by Lmma 3.4 (ii), for " > 0 (to b x latr), thr xists C " such that ~c E ~ ;+ L u " ku u k H + C " + " ku k H + C " : L So ~c E ~ L ;+ u ~c E c ~ ;+ L u + ~c ~ E ;+ (~cu ) L Sic E ~ ;+ ~c L u C " c ku k H + C " c + C k~cu k L " c ku k H + C " c + O : ~c L u O ; 8

19 so a thus T C " ku k H + C " + jt j g c + C " ku k H + C " +. Th trm T ca b writt as T b E ~ ;+ ~a E ~ ;+ u ; u b ~a E ~ ;+ b E ~ ;+ ~a u ; E ~ ; u ; ~ E ; h + b ~a; E ~ ;+i u ; E ~ ; T + T : Similar to th stimats for T, w hav T ~a L u b u L b u u u C max j~a ()j O jj a T C( ~a E ~ L ;+ u + E ~ ;+ ~a L u ) C " ku k H + C0 " + : So Thus jt j C " ku k H + C0 " + R b C ~ P () C ~ u ; u jt j + jt j g c + C " ku k H + C " + C" 0 + : 9

20 Combiig with (3.8), w hav R A z u ; u ( ) h + C 0 ku k H g 0 c 0 h + (C 0 C") ku k H C 4 0; wh is larg ough, C " ku k H + C " + C" 0 C" + C " + by choosig " > 0 a (0; ) such that " C0 C a 8 0h. This ishs th proof of th lmma. To stuy th sstial spctrum of A, w rst look at th Zhisli Spctrum Z A ([9]). A Zhisli squc for A a z C is a squc fu g H, ku k ; supp u fj jj g a A z u! 0 as!. Th st of all z such that a Zhisli squc xists for A a z is ot by Z A. From th abov itio a Lmma 3.5, w raily hav Z A z Cj R z 0 +. (3.) Aothr rlat spctrum is th Wl spctrum W A ([9]). A Wl squc for A a z C is a squc fu g H ; ku k ; u! 0 wakly i L a A z u! 0 as!. Th st W A is all z such that a Wl squc xists for A a z. By ([9, Thorm 0.0]), W A ss A a th bouary of ss A is cotai i W A. So to prov Propositio, it su cs to show that W A Z A. Sic if this is tru, th (3.3) follows from (3.). By ([9, Thorm 0.]), th proof of W A Z A ca b ruc to prov th followig lmma. Lmma 3.6 Giv > 0. Lt C 0 (R) b a cut-o fuctio such that j fjjr0g for som R 0 > 0. D () ; > 0: Th for ach ; A z is compact for som z A, a that thr xists " ()! 0 as! such that for ay u C 0 (R), A ; u " () A u + kuk : (3.) Proof. Sic A N + K, whr N is positiv a is bou, so if z K b ~ C P ~ C : L! L (3.3) k with k > 0 su citly larg, th z A. Th compactss of A + k follows from th local compactss of H,! L. To show (3.), w ot that th graph orm of A is quivalt to kk H. First, w writ K ; b h ~C ; i P ~ C + b ~ C P h ~C ; i : 0

21 W hav h i ~C ; Sic ~ ~ ; E ~ ;+ E ~ ;+ L is bou, so!l ; b 0 () ~ ~E ;+ : h ; D ~ i + ~ D h i ~C ; L!L C a thrfor K ; u C kuk : (3.4) Dot N + a N is th Fourir multiplir oprator with th symbol Th N N N a thus (k) k tah (kh) ( + ik) : (3.5) [N ; ] N [N ; ] + [N ; ] N : Sic [N ; ] 0 () a kn k L!L kn [N ; ] uk C kuk. is bou, w hav To stimat [N ; ], for v C0 (R), w follow [8, p.7-8] to writ Z [N ; ] v () ( y) ( () (y)) v (y) y Z 0 Z 0 Z A v ; () ( y) ( y) 0 ( ( y) + y) v (y) y whr A is th itgral oprator with th krl fuctio K (; y) () ( y) ( y) 0 ( ( y) + y) :

22 Not that () () is th ivrs Fourir trasformatio of i 0 (k) a obviously 0 (k) L, so () L. Thus Z Z Z Z jk (; y)j xy jj ( y) j 0 j ( ( y) + y) y Z Z jj () j 0 j (y) y kk L k 0 k L kk L k 0 k L : So k[n ; ]k L!L C a k[n ; ] N uk L C kn uk L C kuk H : Thus k[n ; ] uk L C + kuk H : Combiig abov with (3.4), w gt th stimat (3.). This ishs th proof of th lmma a thus Propositio. Rcall that to growig mos, w to > 0 such that A has a otrivial krl. W us a cotiuity argumt, by comparig th spctra of A for ar 0 a i ity. First, w stuy th cas ar i ity. Lmma 3.7 Thr xists > 0, such that wh >, A has o igvalus i fzj R z 0g. Proof. Suppos othrwis, th thr xists a squc f g!, a fk g C; fu g H (R), such that R k 0 a A k u 0. Sic A N K M for som costat M ipt of a N is a slf-ajoit positiv oprator, all iscrt igvalus of A li i D M fzj R z M a jim zj Mg : Thrfor, k! k D M with R k 0. Dot () max j~a ()j ; ~ o b () ; j~c ()j ; (3.6) whr ~a () ; ~ b () ; ~c () ar i (3.9) a (3.0). Th ()! 0 as jj!. D th ()-wight L spac L with th orm Z kuk L () juj W ormaliz u by sttig ku k L. W claim that : (3.7) ku k H C, for a costat C ipt of. (3.8)

23 Assumig (3.8), w hav u! u wakly i H. Morovr, u 6 0. To show this, w choos R > 0 larg ough such that max jjr () C. Th Z () ju j C ku k L : jjr Sic u! u strogly i L (fjj Rg), w hav Z Z () ju j lim () ju j jjr! jjr a thus u 6 0. By Lmma 3., A! N strogly i L, thrfor A u! N u wakly. Thus N u k u. Sic R k 0, this is a cotraictio to that N > 0. It rmais to show (3.8). Th proof is quit similar to that of Lmma 3.5, so w oly sktch it. From A k u 0, w hav (N u ; u ) + R b C ~ P () C ~ u ; u R k ku k 0: (3.9) By Lmma 3., (N u ; u ) ( ) h ku k L + C 0 ku k H : Followig th proof of Lmma 3.5, w writ b ~ C P () ~ C u ; u g b E ~ ;+ h g b ~c; E ~ i ;+ + b E ~ ;+ ~a u ; u ; ~ E ; u ; E ~ ; ~ E ; h + b ~a; E ~ i ;+ u ; E ~ ; T + T + T + T : u u u u Th rst trm is stimat as Z T g b Z ju j x b ju j x 3 g c 3 ku k L + C ku k L c ku k L + C ku k L g ku c 3 k L + C ku k L ku c k L + C ku k L g c ku k L + C ku k L ku k L + C ku k L g c ku k L + " ku k L + C " ku k L : 3

24 whr i th sco iquality, w us th fact that jb j ; c 3 ; c C () : 3 Th sco trm is cotroll by T ~c C E ~ ;+ L u + ku k L ku k L C " ku k + ku k H L ku k L C" ku k H + C " ku k L ; whr i th sco iquality w us Lmma 3.4 (iv). Th thir trm is T C ku k L ku k L " ku k L + C " ku k L : By th sam stimat as that of T, w hav T C" ku k H + C " ku k L : Pluggig all of th abov stimats ito (3.9), w hav 0 ( ) g h c ku k L + (C 0 C") ku k H C " ku k L 0 ku k L + C 0 ku k H C " ku k L ; by choosig ; " such that h 0; " C 0 C : Th (3.8) follows. 4 Asymptotic prturbatios ar zro I this Sctio, w stuy th igvalus of oprator A wh is vry small. By Lmma 3., wh! 0+; A! A 0 strogly, whr A 0 N + bp Th rlat oprator i th physical spac is A 0 : H (S )! L (S ) by () : A 0 N + P (x) B b A0 B; which is th strog limit of A wh! 0+. W hav th followig proprtis of A 0. W us A 0 () to ot th pc o th solitary wav paramtr : 4

25 Lmma 4. (i) Th oprator A 0 : H (R)! L (R) is slf-ajoit a ss A 0 [ h g c ; +): (ii) x () kr A 0 a A 0 has at last o gativ igvalu that is simpl. (iii)ur th hypothsis (H) of o scoary bifurcatio, kr A 0 () f x ()g wh is ot a turig poit a kr A 0 () x () wh is a turig poit. For ay > 6, x () is th oly o krl of A 0 (). (iv) Wh 6 is small ough, A0 () has xactly o gativ igvalu a kr A 0 () f x ()g. Ur hypothsis (H), th sam is tru for A 0 () with 6 ;, whr is th rst turig poit. (v) Wh!, th umbr of gativ igvalus of A 0 () icrass without bou. Proof. (i) Th sstial spctrum bou follows from th obsrvatios that ss (N ) [ h ; +) a bp! g c wh jj!. Proof of (ii): To show x () kr A 0 ; it is quivalt to show that O S ; w hav a So P x (x) x (x; (x)) kr A 0 : x (x) + x (x) 0; P x (x) + x P (x) 0 (4.) P x (x) ( x xx + x ) ( x y + x ) (4.) x (x) x ( ) x ( x) : P x P x (x) x ( x) N ( x (x)) ; a thus A 0 x (x) 0. Now w show that A 0 has a gativ igvalu. W ot that th Fourir multiplir oprator N h has th sam symbol as i th Itrmiat Log Wav quatio (IIW), for which it was show i [] that for K > 0 larg, th oprator (N +K) is positivity prsrvig. Thus, by th spctrum thory for positivity prsrvig oprators ([]), th lowst igvalu of A 0 is simpl with th corrspoig igfuctio of o sig. Sic x (x) is o, x () has a zro at 0. So 0 is ot th lowst igvalu of A 0 a A 0 has at last o simpl gativ igvalu. To prov (iii)-(iv), rst w show that th oprator A 0 () is xactly th oprator A () itrouc by Plotikov ([58, p. 349]) i th stuy of th bifurcatio o 5

26 of solitary wavs. I [58]; h is st to a th paramtr is th F () ivrs squar of th Frou umbr, th th oprator A () is by A () N a; a () xp (3) cos + 0 () I th abov, xp ( + i) W whr W is by (.8), as ca b s from ([58, (4.), p. 348]) with u w. To show that A 0 () A ( ()), it su cs to prov that A () x () 0: (4.3) Sic this implis that 0 A () A 0 () x () a bp x () a thus bp a. W prov (4.3) blow. I [58], solitary wavs ar show to b critical poits of th fuctioal J (; w) Z wn w w ( + N w) : (4.4) R Lt th slf-ajoit oprator A 0 () to b th sco rivativ of J (; w) at a solitary wav solutio. I [58, p. 349]; th oprator A () is via A () M A 0 () M: Hr, th oprator M : L! L is by Mf f ( + Cw 0 ) + w 0 Cf R fw Rfg ; (4.5) whr C is i Sctio such that Rf f icf is th bouary valu o f& 0g of a aalytic fuctio o D 0. Our itio (4.5) abov aapts th otatios i [4, p. 8], which stuis th bifurcatio of Stoks wavs by usig a similar variatioal sttig as [58]. Takig of th quatio r w J (; w) 0 for a solitary wav solutio w, w hav A 0 () w 0 0. Sic Rw M w 0 0 w 0 icw 0 R R W + Cw 0 + iw 0 R ( (w 0 icw 0 ) ( + Cw 0 iw 0 ) jw j w0 jw j c v c x () ; ) w hav M x () cw 0 a thus A () x () cm A 0 () w 0 0. This ishs th proof that A 0 () A (). 6

27 Proof of (iii): By applyig th aalytic bifurcatio thory i [4], [5] to th variatioal sttig (4.4) for th solitary wavs, o ca rlat th scoary bifurcatio of solitary wavs with th ull spac of A 0 (quivaltly r wwj ):Ur th hypothsis (H), thr is o scoary bifurcatio a thrfor th krl of A 0 is ithr u to th trivial traslatio symmtry ( x ) or u to th loss of mootoicity of () at a turig poit which grats a aitioal I th Appix, w prov that at a turig poit 0, A 0. By [3] thr is o asymmtric bifurcatio for solitary wavs with F >, so x () is th oly o krl of A 0 (). Proof of (iv): Lt () xp 3, th 6 is quivalt to a thus is a small paramtr. Cosir a igvalu of A 0 ( ()), lt (3 ()). By usig th KDV scalig, it was show i [58, p. 353] that wh! 0, th limit (0) is a igvalu of th oprator B 3 3 x + 9 sch x which has thr igvalus 3 4 ; 3 a 7 4. Thrfor, wh is small, A0 has thr 5 igvalus 4 + o ; o a o. Sic 0 is a igvalu of A 0, th mil o must b zro a th rst two igvalus ar o positiv a o gativ. Ur th hypothsis (H), wh <, that is, bfor th rst turig poit, w hav kr A 0 () f x ()g. Th for all 6 ; ; th oprator A 0 () always has oly o gativ igvalu. Suppos othrwis, th wh icrass from 6 to, th igvalus of A 0 () must go across zro at som 6 ; :This implis that im kr A 0 ( ), a cotraictio to (H). Proprty (v) is Thorm 4.3 i [58]: W ot that by Lmma 4. (iv), thr is o scoary bifurcatio for small solitary wavs. Although this fact was ot stat xplicitly i [58], it coms as a corollary of rsults thr. Nxt, w stuy th igvalus of A for small. Sic th covrgc of A! A 0 is rathr wak, w caot us th rgular prturbatio thory. W us th asymptotic prturbatio thory vlop by Vock a Huzikr ([66]), s also [9], [30]. First, w stablish som prlimiary lmmas. Lmma 4. Giv F C 0 (R). Cosir ay squc! 0+ a fu g H (R) satisfyig A u + ku k M < (4.6) for som costat M. Th if w lim! u 0, w hav lim kf u k! 0 (4.7) a lim! A ; F u 0: (4.8) 7

28 Proof. Sic (4.6) implis that ku k H C, (4.7) follows from th local compactss of H,! L. For th proof of (4.8), w us th sam otatios as i th proof of Lmma 3.6. W writ A N + K. Th [N ; F ] [N N ; F ] N [N ; F ] + [N ; F ] N ; whr N + a N has th symbol (k) by (3.5). W hav k[n ; F ] u k kf 0 u k! 0, agai by th local compactss. Sic k (k)! 0 wh jkj!, by [8, Thorm C] th commutator [N ; F ] : L! L is compact. This ca also b s from th proof of Lmma 3.6, sic [N ; F ] R 0 A whr A is a itgral oprator with a L (R R) krl: Sic kn u k ku k H C a u! 0 wakly i L, w hav v N u! 0 wakly i L. Thrfor, k[n ; F ] N u k k[n ; F ] v k! 0. Thus, k[n ; F ] u k! 0. Sic K ; F u b E ~ ;+ P h b F; E ~ ;+i P p + q : ~ E ;+ ~ E ;+ ; F u + b ~ E ;+ P For ay " > 0, by Lmma 3.4 (iii), wh is larg w hav F E ~ ;+ L u " ku k H " ku k H "M : h F; ~ E ;+i u So F kq k C E ~ ;+ u + kf u k "CM + C kf u k "CM ; wh is larg. By th sam proof as that of (3.5), for ay > 0; w hav th stimat E ~ ;+ C (ipt of ). H!H Dot r P ~ E ;+ th kr k H C ku k H CM. Sic w lim! u 0, w hav w lim! r 0 as i th proof of Lmma 3.4 (iii). Th similar to th stimat of kq k, w hav F kp k C E ~ ;+ r + kf r k "CM ; wh is larg. Thrfor, K ; F u 4"CM wh is larg ough. Sic " is arbitrary, w hav K ; F u! 0; wh!. This ishs th proof of (4.8). u ; 8

29 Lmma 4.3 Lt z C with R z 0, th for som > 0 a all u C0 (jxj ), w hav A z u 4 0 kuk ; (4.9) wh is su citly small. Hr 0 > 0 is by (3.4). Proof. Th stimat (4.9) follows from R A z u; u 4 0 kuk : (4.0) Th proof of (4.0) is almost th sam as that of Lmma 3.5, xcpt that Lmma 3.4 (iii) is us i th stimats. So w skip it. With abov two lmmas, w ca us th asymptotic prturbatio thory ([9], [30]) to gt th followig rsult o igvalu prturbatios of A. Propositio Each iscrt igvalu k 0 of A 0 with k 0 0 is stabl with rspct to th family A i th followig ss: thr xists ; > 0, such that for 0 < <, w hav (i) B (k 0 ; ) fzj 0 < jz k 0 j < g P A ; whr P A zj R (z) A z o xists a is uiformly bou for (0; ) : (ii) Dot I P fjz I R (z) z a P 0 k 0jg fjz R 0 (z) z k 0jg to b th prturb a uprturb spctral projctio. Th im P im P 0 a lim!0 kp P 0 k 0: It follows from abov that for small, th oprators A hav iscrt igvalus isi B (k 0 ; ) with th total algbraic multiplicity qual to that of k 0. 5 Movig krl a proof of mai rsults To stuy growig mos, w to ursta how th zro igvalu of A 0 is prturb, i particular its movig irctio. I this Sctio, w riv a movig krl formula a us it to prov th mai rsults. W assum hypothsis (H) a that is ot a turig poit. Th by Lmma 4. (iii), kr A 0 () f x ()g. Lt ; > 0 b as giv i Propositio for k 0 0. Sic im P im P 0, wh < thr is oly o ral igvalu of A isi B (0; ), which w ot by k R. Th followig lmma trmis th sig of k wh is su citly small. 9

30 Lmma 5. Assum hypothsis (H) a that is ot a turig poit. For > 0 small ough, lt k R to b th igvalu of A ar zro. Th k lim!0+ whr E () is th total rgy i (.8): Th followig a priori stimat is us i th proof. c E c k xk L ; (5.) Lmma 5. For > 0 small ough, cosir u H (R) satisfyig th quatio A z u v, whr z C with R z 0 a v L. Th w hav kuk H C kuk L + kvk L ; (5.) for som costat C ipt of. Hr, th orm kk L is i (3.7) with th wight () by (3.6). Proof. Th proof is almost th sam as that of th stimat (3.8) i th proof of Lmma 3.7. So w oly sktch it. W hav (N u; u) + R b C ~ P () C ~ u; u R z kuk R (u; v) : By th sam stimats as i provig (3.8), xcpt that Lmma 3.4 (iii) is us, w hav ( ) h g c 0 kuk L + (C 0 C") kuk H C " kuk L " kuk L + " kvk L : Th th stimat (5.) follows by choosig " > 0 a (0; ) proprly. Assumig Lmma 5., w prov Thorm. Proof of Thorm. W x (~ ; ). Ur th assumptio (H), it follows from Lmma 4. that A 0 () has oly o gativ igvalu k 0 < 0 a kr A 0 () f x ()g. By Propositio a Lmma 5., thr xists ; > 0 small ough, such that for 0 < <, A has o gativ igvalu k i B k 0 ; with multiplicity a o positiv igvalu k i B (0; ) bcaus E c E0 () c 0 () < 0 for (~ ; ). Cosir th rgio fzj 0 > R z > M a jim zj < Mg ; whr M is th uiform bou of A N. W claim that: for small ough, A has xactly igvalus (coutig multiplicity) i fzj > R z > M a jim zj < Mg : That is, all igvalus of A with ral parts o gratr tha li i B k 0 ; [ B (0; ). Suppos othrwis, thr xists a squc! 0+ a fu g H (R) ; z B k 0 ; [ B (0; ) 30

31 such that A z u 0. W ormaliz u by sttig ku k L. Th by Lmma 5., w hav ku k H C. By th sam argumt as i th proof of Lmma 3.7, u! u 6 0 wakly i H. Lt z! z B k 0 ; [ B (0; ) ; th A 0 u z u which is a cotraictio. This provs th claim. Thus for small ough, A has xactly o igvalu i. Suppos th coclusio of Thorm os ot hol, th A () has o krl for ay > 0. D () to b th umbr of igvalus (coutig multiplicity) of A i. By (3.3), th rgio is away from th sstial spctrum of A, so () is a it itgr. For small ough, w hav prov that (). By Lmma 3.7, () 0 for >. D th two sts S o f > 0j () is og a S v f > 0j () is vg : Th both sts ar o-mpty. Blow, w show that both S o a S v ar op. Lt 0 S o a ot k ; ; k l (l ( 0 )) to b all istict igvalus of A 0 i. Dot ih ; ; ih m to b all igvalus of A 0 o th imagiary axis. Th jh j j M, j m. Choos > 0 su citly small such that th isks B (k i ; ) ( i l) a B (ih j ; ) ( j m) ar isjoit, B (k i ; ) a B (ih j ; ) os ot cotai 0. Not that A ps o aalytically i (0; +). By th aalytic prturbatio thory ([9]), if j 0 j is su citly small, ay igvalu of A i lis i o of th isks B (k i ; ) or B (ih j ; ). So () is ( 0 ) plus th umbr of igvalus i [ m i B (ih j; ) with th gativ ral part. Th latr umbr must b v, sic th complx igvalus of A appars i cojugat pairs. Thus, () is o for j 0 j small ough. This shows that S o is op. For th sam raso, S v is op:thus, (0; +) is th uio of two o-mpty, isjoit op sts S o a S v. A cotraictio. So thr xists > 0 a 0 6 u H (R) such that A u 0. Lt f B u H (S ), th A f 0. D (x) C f; P (x) P (x) a (x; y) to b th solutio of th Dirichlt problm 0 i D ; j S f ; (x; h) 0: Th ( (x) ; (x; y)) satis s th systm (.3)-(.7), thus t [ (x) ; (x; y)] is a growig mo solutio to th liariz problm (.). Blow w prov th rgularity of [ (x) ; (x; y)]. Sic th oprator C is rgularity prsrvig, from A f 0 w hav (x) C P C f H (S ) : By th lliptic rgularity of Numa problms ([]), w hav (x; y) H 5 (D ) : So by th trac thorm, f j S H (S ). Rpatig this procss, w gt (x) H (S ) a (x; y) H 7 (D ) : Sic th irrotatioal solitary wav pro l a th bouary S ar aalytic ([36]), w ca rpat th abov procss 3

32 to show (x; y) H k (D ) for ay k > 0. Thrfor (x) C ( j S ) H k (S ), for ay k > 0. By Sobolv mbig, [ (x) ; (x; y)] C. This ishs th proof of Thorm. Proof of Thorm. Lt < ; < < b all th turig poits. Th! +: Ur th assumptio (H), kr A 0 () f x ()g, for ( i ; i+ ) ; i. Dot by () th umbr of gativ igvalus of A 0 (). Th () is a costat i ( i ; i+ ), by th sam argumt i th proof of Lmma 4. (iv). Dot ~ < ~ ; < ~ < to b all th critical poits of E (). Each ~ k lis i som itrval ( i ; i+ ). Th th sig of Ec E 0 () c 0 () chags at ~ k i ( i ; i+ ). So w ca a itrval I k ( i ; i+ ) such that th umbr ~ () () + ( + sig (E 0 () c 0 ())) is o for I k. Not that ~ () is th umbr of igvalus of A () i th lft half pla, for su citly small. So by th sam proof as that of Thorm, w gt a purly growig mo for solitary wavs with I k. Sic ~! ; th itrvals I gos to i ity. W mak two rmarks about th ustabl solitary wavs prov abov. Rmark I trms of th paramtr! F qc, it was fou ([63]) from umrical computatios that th rgy maximum is achiv at! 0:88, which corrspos to th amplitu-to-pth ratio (0) h 0:784 ([5]). Th highst wav has th paramtrs!, 0:833 a th maximal travl sp is achiv at! 0:97; 0:790 ([63], [48]). So th ustabl wavs prov i Thorm a ar of larg amplitu, a thir hight is comparabl to th watr pth. Thrfor, this typ of istability caot b captur i th approximat mols bas o small amplitu assumptios. I, although som approximat mols shar crtai faturs of th full watr wav mol, o ustabl solitary wavs hav b fou. For xampl, th Gr-Naghi mol is propos to mol watr wavs of largr amplitu a it also has a i it rgy fuctioal, but th umrical computatio ([37, P. 59]) iicats that all th G-N solitary wavs ar spctrally stabl. For Camassa-Holm a Dgaspris- Procsi quatios, thr xist corr solitary wavs (pakos). Howvr, ths pakos ar show to b oliarly stabl ([], [46]). So th istability of larg solitary wavs sms to b a particular fatur of th full watr wav mol. Rmark Th liar istability suggsts that th solitary wav caot prsrv its shap for all th tim. Th log tim volutio arou a ustabl wav was stui umrically i [64]. It was fou that small prturbatios with th sam amplitu but opposit sigs ca la to totally i rt log tim bhaviors. For o sig, th prturb wav braks quickly a for th othr sig, th prturb wav vr braks a it ally approachs a slightly lowr stabl solitary wav with almost th sam rgy. Not that i th brakig cas, th iitial prturb pro l has a rathr gativ slop ([64]). Th wav brakig for shallow watr 3

33 wavs mols such as Camassa-Holm ([]) a Whitham quatios ([60], [54]) is u to th iitial larg gativ slop. It woul b itrstig to clarify whthr or ot th wav brakig fou i [64] has th sam mchaism. Th wav brakig u to th istability of larg solitary wavs ha b us to xplai th brakig wavs approachig bachs ([5], [56], [57]). Wh a wav approachs th bach, th amplitu-to-pth ratio ca icras to b ar th critical ratio ( 0:784) for istability a cosqutly th wav brakig ca occur. It rmais to prov th movig krl formula (5.). Proof of Lmma 5.. As scrib at th bgiig of this Sctio, for > 0 small ough, thr xists u H (R), such that A k u 0 with k R a lim!0+ k 0. W ormaliz u by ku k L. Th by Lmma 5., w hav ku k H C a as i th proof of Lmma 3.7, u! u wakly i H. Sic A 0 u 0 0 a kr A 0 () f x ()g, w hav u 0 c 0 x () for som c Morovr, w hav ku u 0 k H 0. To show this, rst w ot that ku u 0 k L! 0, sic ku u 0 k L Z jjr () ju u 0 j + max jjr () ku u 0 k L ; a th sco trm is arbitrarily small for larg R whil th rst trm ts to zro by th local compactss. Sic A k (u u 0 ) k u 0 + A 0 A u 0 ; by Lmma 5. w hav ku u 0 k H C ku u 0 k L + jk j ku k L + A 0 A u 0 L! 0; wh! 0+. W ca st c 0 by rormalizig th squc. Nxt, w show that lim!0+ k 0. From A k u 0; w hav A 0 u + A A 0 u k u : (5.3) Takig th ir prouct of abov with x (), w hav k (u ; x ()) A A 0 u ; x () : m () : W comput th itgral m () i th physical spac, by th chag of variabl! x. Dot by h ; i th ir prouct i L (S ). Notig that x b (), 33

34 w hav A m () A 0 + u (x) ; x (x) P P u ; x P E ;+ u ; x m + m + m 3 ; u ; x whr w us A A 0 C P (x) C P P P P (x) + P E ;+ : P (5.4) W comput ach trm sparatly. For th rst trm, wh! 0+; P P m () u ; x u ; D x P u ; E ; P! x; 0; whr w us Lmma. (b) i th abov a th rsultat itgral is zro bcaus P ; ; ar v a x is o i x. Th sco trm is m ()! P u ; x D x (u ) u 0; u ; x; u ; E ; u ; x ( x) u whr th rlatios (4.) a (4.) ar us i th abov computatio. For th last trm, P m 3 () E ;+ u ; x P E ;+ u ; D x P E ;+ u ; E ;! 0; 34

35 bcaus E ;+ u! 0 wakly i L, wh! 0+. So m ()! 0, a thus k lim!0+ lim m ()!0+ (u ; x ()) 0: Now w writ u c x + v, with c (u ; x ) ( x ; x ). Th (v ; x ) 0 a c! as! 0+. W claim that: kv k L C (ipt of ). Suppos othrwis, thr xists a squc! 0+ such that kv k L. Dot ~v v kv k L. Th k~v k L a ~v satis s A ~v k~v k L k u c A A 0 x () : (5.5) Dot whr g () A A 0 x () b () w (x ()) ; w (x) A A 0 x (x) (5.6) P + E ;+ P x P x (u ) + P E ;+ u + P x E ;+ E ;+ P x E ;+ P By Lmma., kw k L (S ) C (ipt of ), a morovr w (x)! u + P E ;+ : strogly i L (S ); wh! 0 +. (5.7) So kg k L C a thus by applyig th stimat (5.) to (5.5), w hav k~v k H C. Thrfor, as bfor, ~v! ~v wakly i H. Sic k ; k~v k L! 0, w hav A 0 ~v 0 0. So ~v 0 c x () for som c 6 0. But (~v ; x ()) 0 implis that (~v 0 ; x ()) 0; a cotraictio. This provs that kv k L C. Th quatio satis by v is A v k A A 0 u c x () k u c b () w (x ()) : Applyig Lmma 5. to th abov quatio, w hav kv k H v! v 0 wakly i H. By (5.7), v 0 satis s A 0 u v 0 b () + P () : 35 C a thus

36 It is show i th Appix that A c (x) u + P (x) ; (5.8) a quivaltly, A c () u b () + P () ; whr (x; y) (x; y) So A 0 v c () 0. Sic (v 0 ; x ()) lim!0+ (v ; x ()) 0; w hav v c () + 0 x () ; c ; x k x k L : By th sam argumt as i th proof of ku u 0 k H! 0, w hav kv v 0 k H! 0: W rwrit u c x + v c x + v ; whr c c + 0, v v 0 x. Th c!, c () wh! 0+. k Now w comput lim!0+. From (5.3), w hav A 0 u + A A 0 c cy: x + v k u : Takig th ir prouct of abov with x (), w hav k A (u A 0 A A 0 ; x ()) c x ; x + v ; x c I + I : Agai, w o th computatios i th physical spac. For th rst trm, w us (5.6) to gt A I A 0 w (x) x (x) ; x (x) D u P D ; x (x) + ( ) ( ) P E ;+ ; x (x) I + I + I 3. ; x (x) ; x (x) 36

37 W hav u I hu ; x i + u ; D x ; x u u ; E ; ; x! u u ; ; x ; P I! ; u a I 3 P ; x (u ) ; ; x ; D P x (u ) ; x ; E ; E ;+ ; x P E ;+ ; P E ;+ ; E ;! P ; : u D x So To comput I, w writ P I I + I + I 3. lim I ()!0+ ; u P v ; x + P E ;+ v ; x + P : v ; x W hav I P v ; D x P v ; E ;! c ; ; I v ; v ; + E ; u x (u )! v ; c ; u x (u ) 37

38 a I 3 P E ;+ v ; x P E ;+ v ; P E ;+ v ; + E ;! 0: D x So Thus lim I c ; u + P : k lim!0+ P c lim!0+ c I + I (u ; x ) D ; u It is show i th Appix that ; u E + P + P D c ; u + P k x k : c ; u whr th momtum P is by Z P S x (x; (x)) x; + P ; (5.9) with cx. It is show i [0] (s also [47]) that for a solitary wav solutio E c cp c ; whr w ot that th travl irctio cosir i this papr is opposit to th o i th abov rfrcs. A combiatio of abov rsults yils th formula (5.). As a corollary of th abov proof, w show Thorm 3. Proof of Thorm 3. Th proof is vry similar to that of Lmma 5., so w oly sktch it. Th mai i rc is that th computatios p o th paramtr. W us ; ; ; tc. to ot th pc o, a ; tc. for quatitis pig o 0. Dot u () B j S;. Th A u 0 a w ormaliz u by ku k L. First, w show that! 0 strogly i L (R). I, for ay v L (R), sic th orms ~E ; kk L a kk L b ; ar quivalt, w hav E ~ ; v L C E ~ ; v L b ; Z C + k ~ M ; vk L b ; Z C kvk L + C k M ~ ; vk L A b + A ; ; b ; jj 38

39 o whr > 0 is arbitrary a ~M; ; R is th spctral masur of th slf-ajoit oprator R ~ i D ~ o L b. For ay " > 0, sic M ; f0g 0 w ca choos small ough such that Z k M ~ vk L " b jj : Sic th masur k M ~ ; vk covrgs to k M ~ L vk, w hav A L b ; b " wh is big ough. Th trm A ts to zro wh! : Bcaus " ca b arbitrarily small, w hav lim E ~ ; v 0: L! So A! A 0 strogly a similar to th proof of Lmma 5., w hav u! x () i H (R) by a rormalizatio. W writ u c x; () + v ;with c (u ; x; ) ( x; ; x; ). Th c! a (v ; x; ) 0. As bfor, w hav v! v 0 i H (R), whr W rwrit v c () + 0 x () ; c ; x ( x ; x ) : whr c c + 0 a v v!. Similarly, w hav A 0 c u c x; () + v c x; + v ; A 0 A x;; x; + 0 x;. Th c!, c () wh A 0 v ; x; c I + I : As bfor, th calculatios of I ; I ar rst o i th physical spac S ; with th ir prouct h:; :i. W us th sam otatios as i th proof of Lmma 5.. By th sam computatios, w hav I! u ; ; bu () ; + E ; ; () ; u ; b u ; () ; : + ~ E ; ; ; () Th othr trms ar hal similarly. So ally w hav ; u 0 lim! c I + I c E c j 0 : + c ; u + P So E 0 ( 0 ) 0. This ishs th proof of Thorm 3. 39