LECTURE 4. Solitonic lattices, overlapping and embedded solitons. Applications of finite-band quasi-periduc potentials.

Transcription

1 LCTUR 4. Solto lattes, ovela ad ebedded soltos. Alatos of fte-bad quas-edu otetals.

2 Goud State of a doed Peels sulato : Nea the half fll - ouble Coesuablty Pot Pobles that all fo a seal vestato of the eo ρ : states wth ρ = ad wth ρ - ~ oesuate CW ae desbed by dffeet odels, hav dffeet eleto oetes. The hooeeous state = ost the ole-feld odel s stoly hooeeous fo the ot of vew of the odel of the eal feld. B A s a Itoduto of addtoal eletos to the state ρ = leads to foato of soltos wth a loal level ly at the ete of the fobdde bad. As ρ the defoato should take the fo of a solto latte that oes ove to the susodal stutue. The da state of the solto ves se to a ew allowed bad loated syetally elatve to the ete of the etal bad = A

3 Peod soluto: the latte of altude soltos - Solto latte of the ode aaete, esty of d-a states s dstbuted ea the a zeos. If elted, eah eleet beoes a atle - Altude Solto

4 Bad stutue at esee of the solto latte New Bllou zoe boudaes oesod to eodty of the solto latte Oly oe fte ube eeal fobdde a Cotaly to oo eetatos of the bad theoy that as oes at ay eetto of the Ba veto ᴂ

5 W d a Peels state esults fo sotaeous syety beak fo the teato of the etal eletos wth the latte defoatos. I sulato of the obed tye, = e + e e =st s odued by the bas stutue of the ha s the otbuto ade by the defoato of the latte stablzed by the teato wth the eletos, ad =st s the hase dffeee betwee the at eleets of the teat eletos wth otetals v se to e ad

6 Iteals of otos I of the KdV aea the sees easo of the oetu ove owes of / : I o I d ao I I Ud I U d U P, Q defoato eey otas addtoal te I, whh s ot eeal a KdV teal. Nevetheless, the sle-bad ase the futoal I s a KdV teal. I ths ase P ad Q satsfy the statoay KdV eq., Δ satsfes the statoay KdV eq.. I the Haltoa foulato I I I P t t I Fo ths equato, di/dt =,.e., I s a MKdV ad a KdV teal

7 * * d z e e e s e e u v e ey futoal os / v v v u u u Shoede eqs. fo ooets a eqs. P u P Q v Q u v * F ve ve F 4 Q F Q F

8 os d F d e The fst te wth e oes fo the te the syety beak te e oated the te the eey futoal evous ae, also Letue whh does ot belo to the faly of oseved teals. Stll be the azats woks susly: The self-osstey odto s a futoal equato fo the set of futos F ad Δ. We stulate that t be eduble to a aleba equato. I the slest ase ths eas that fo eah d B d F A elude F fo ths equato ad obta a sle equato fo the futo Δ : 6 ' 4 A B A B C U 6UU au, U P, Q U' U au bu

9 ' bq aq Q Q Q R R s, ; : ; : G Z ; the setu of the Shoede equato wth a otetal satsfy KdV has oly oe fobdde bad

10 leto wave futos at esee of solto latte y dy R A v e d R L L L d A L

11 K - the fst elato betwee the thee fee aaetes,,. Two othe elatos a be obtaed fo the self-osstey odtos: J J J J k J e os R J A A quatos otas loathally dve sus. We eulaze the by subtat a sla equato at the uetubed state ; ~ e / v F

12 We subtated the eey of the sae syste at =. The oeto betwee the utoff oetu ad the utoff eey a be obtaed, e.., fo the bae Shoede eq. d R d L W W e e / os os K P The total eey a be alulated oly by tak aout the shft of eey-oetu elato at the lowest utoff. Relato to hase shfts fo the sle solto, letue.

13 The otetals P ad the futos u ae defed by sla eessos but wth Δ -Δ ad allow fo the shft + Se both as ae oated by the sae futo Δ, we a detee : v v u u k F as, k s k, k,

14 ,,,, k s k s R K k s k s k K e

15 whh oesods to a doa stutue a syste wth doubly deeeate oud state At e the otooal ase =/ s sled out s,

16 Os of ult-eodtes ad of soltos at the eodtes bah. Fo half fll total ube of atles = + = I ρ I /a S olazato = - Zeea sltt Fo botto a tal value H suh fo the sotaeous olazato to eee, as futo of ρ

17 Stateets of Novkov theoy fo seta.. Fo a Shoede, a, Toda, et. eqs. fo eevalues, thee est ult eod otetals suh that the ube N of allowed ad fobdde as bads s fte. All the est follows fo ths stateet!. The ae takes lae at the Rea sufae the ole lae of the eey : R whee ae the bads s.. Oe eodty s assoated wth eah a. Peodtes ae ot loked, the elatve dslaeets, whle adjust the shaes, take o eey hee the zeo odes. Whe a a ollases + the the oesod eodty s deeeated to a sle solto. The solto shae ves the efletoless otetal fo the Haltoa.. The otetals satsfy NLPs fo the heahy of: KdV fo Shoede, KdV fo the eal feld a, NLS fo the ole feld a. t u=δi/δu whee Iu, u s a lea obato of fst N fo the fte sees of teals of oto I. evalues also fo aothe sees of teals of oto.. Futoal eqs. fo a eteu of ay lea obato of k ad I ae edued to aleba oetos. Paaetes ae the teals ove loos fo the tous of the Rea sufae. Key to ou oals.

18 Solto uo a eod stutue: two lt ees. Solto ebedded to a ae aay of waves Sufes hut: Ovela solto odulates sequee of waves. Whateve ee o solto s osto, the aay shft aoss the solto s always half of the eod.the solto s tooloally otval but s ot tooloally stable. It s stable eeetally aoodat the uaed s ½ at the slt off level. The hae s vaable: evolves fo e to zeo.

19 oubly Peod Suestutues, S tatos, Oe-leto States we osde, aast the bakoud of the eod stutue, soltos that ay loalzed eleto states. These soltos oesod to states of the syste wth oe eta eleto. The tlet ad slet etatos ae sus of two sle-atle states. We fd that the solto hae. otast to the s, s atally seeed at ρ ; t evolves fo e at half-fll ad vashes adually the oesuate lt I the Fohlh lt at F the loal hae q vashes lke q e F F a qualtatve hae takes lae the fo of the solto, fo a solated olao to a evelo solto aast the bakoud of a adly osllat stutue

20 W, f f, d

21 two futos, aodae wth the ube of fobdde bads whh ae defed the eos of the fobdde bads G, : Q satsfes the seod KdV equato R R d LA v e q l d d R Ld Ld dn 4 R LA d dn R 5 C C Q Q QQ Q Q

22 Oly of B s ae deedet, we ave to self-osstey equatos ad the elato, A A A f L B B B 6 we have B B B =B =B = 4 Fo B B d d sle eod otetal wth oe fobdde bad Fo double eod otetal we have to ut

23 Two oe teals satsfy the ve oetato ad olazato d d d ; d d d addtoal odtos esues equalty of the oeta at the oleetay edes of fobdde bads aalytty o the Rea sufae. G dn

24 Just as the ase of a sle-bad otetal ad the lt of a solated solto, the total eey of the syste s deloalzed,.e., The suato dvees adly fa fo the Fe levels. We eulaze t by subtat the eey desty of the Peels delet W /L at ρ = Se the su ove ovees fo w w, we a alulate w ad w seaately by todu a lt oetu whh should be fed ode that the haes of the atle destes be oeted wth the oeta ± s oeted wth the eey oesod to ths state ost L W w L W w, * f w A L w O Q, O ost Q

25 R d R d w ost Q w w,,, w w

26 At /v F the solto hae s lose to the oe-eleto hae. Wth eas, atal see of the solto takes lae o aout of defoato of the eod stutue the vty of the solto. I the Fohlh lt /v F q s,.e., the hae s oletely seeed. Resdual hae s a weak effet of a dstat oesuablty ot We osde the lt of a ftely sail s desty, whe the ube of states, ad the ube of atles that s equal to t, the bad {-,} teds to zeo. The bad tself otats ths ase to a loal level ; F F F F S v v v v e K F e q l 4, / Sle-leto States ad S tatos

27 e e dt d H e e v v v v v L,, A,, v v H a N I l N I v L A 8 Toda: oesuate doubly eod stutue

28 Todda latte, quas-eod soluto: oesuate doubly eod stutue -- a sueosto of two sublattes wth eve ad odd, shfted elatve to eah othe, ad wth eods ellt futo Fo a solated kk 9 tah / osh / osh l u u ost / / l / / l u, 4 / tah / tah e e

29 * ˆ H F v d W * * * * * * *,, * * * * ost J L d L d L I d L I * * 8 I I ICW wth dseso of the oul ostat. Peod soluto.

30 eeeay fo the te evoluto: ost e l