a Xi :2006.00958 1 [nlin.PS] 1 Jun 2020
Kuzne so -Ma b ea he -like solu ions in he Sale no model
J. Sulli an,1, ∗E. G. Cha alampidis,2, †J. Cue as-Ma a e ,3, ‡P. G. Ke ekidis,1, 4, §and N. I. Ka achalios5, ¶
1Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s Amhe s , Amhe s , MA 01003-4515, USA
2Ma hema ics Depa men , Cali o nia Poly echnic S a e Uni e si y, San Luis Obispo, CA 93407-0403, USA
3G upo de F´ısica No Lineal, Depa amen o de F´ısica Aplicada I,
Uni e sidad de Se illa. Escuela Poli ´ecnica Supe io , C/ Vi gen de ´
A ica, 7, 41011-Se illa, Spain
Ins i u o de Ma em´a icas de la Uni e sidad de Se illa (IMUS). Edi icio Celes ino Mu is. A da. Reina Me cedes s/n, 41012-Se illa, Spain
4Ma hema ical Ins i u e, Uni e si y o Ox o d, Ox o d, UK
5Depa men o Ma hema ics, Uni e si y o he Aegean, Ka lo assi, 83200 Samos, G eece
(Da ed: June 2, 2020)
The Sale no model is a disc e e a ian o he celeb a ed nonlinea Sch ¨odinge (NLS) equa ion in e pola ing
be ween he disc e e NLS (DNLS) equa ion and comple ely in eg able Ablowi z-Ladik (AL) model by app o-
p ia ely uning he ele an homo opy pa ame e . Al hough he AL model possesses an explici ime-pe iodic
solu ion known as he Kuzne so -Ma (KM) b ea he , he exis ence o ime-pe iodic solu ions away om he
in eg able limi has no been s udied as o ye . I is hus he pu pose o his wo k o shed ligh on he exis ence
and s abili y o ime-pe iodic solu ions o he Sale no model. In pa icula , we a y he homo opy pa ame e o
he model by employing a pseudo-a cleng h con inua ion algo i hm whe e ime-pe iodic solu ions a e iden i ied
ia ixed-poin i e a ions. We show ha he solu ions ans o m in o ime-pe iodic pa e ns ea u ing small, ye
non-decaying a - ield oscilla ions. Rema kably, ou nume ical esul s suppo he exis ence o p e iously un-
known ime-pe iodic solu ions e en a he in eg able case whose s abili y is explo ed by using Floque heo y.
A con inua ion o hese pa e ns owa ds he DNLS limi is also discussed.
PACS numbe s:
I. INTRODUCTION
The s udy o ogue wa e pa e ns has been a ocal poin o ecen esea ch in dispe si e nonlinea wa e sys ems [1–7]. Whe he
hey may a ise “spon aneously ou o nowhe e and disappea wi hou a ace” [8], o gene a ed g adually h ough ene gy ans e
in mul iple soli on collision, hey a e s udied in ensely in a di e se ange o se ings [9–12]. Rele an s udies ha e appea ed
om supe luid helium [13], o plasmas [14] and om nonlinea op ics [15–21] o he, a guably, mos na u al enue o wa e
wa es [22–25].
In he disc e e ealm, he e is a ewe s udies. This is in a sense ela i ely na u al o expec . Much o wha we know abou
ogue wa es is in ima ely connec ed o echniques s emming om in eg able sys ems and he disc e e se ing o di e en ial-
di e ence equa ions is no excep ion. The mos p o o ypical model associa ed wi h in eg abili y he e is he so-called Ablowi z-
Ladik (AL) model [26, 27]. In his se ing, ogue wa es in he o m o he p o o ypical Pe eg ine soli on [28], bu also in he o m
o he Kuzne so [29], Ma [30], and Akhmedie [31] b ea he s ha e been iden i ied in he wo k o [32]. The Pe eg ine soli on
is a solu ion ha is localized in bo h space and ime and i is a special membe o a amily ex ending be ween he spa ially
pe iodic Akhmedie solu ions and he empo ally-pe iodic Kuzne so -Ma (KM) ones. In eg abili y has also p o ided highe
o de solu ions [33]. Howe e , beyond hese indings he e is e y li le. A ques ion ha is na u al o ask is wha o his s uc u e
pe sis s in he mo e physically ele an ( o se ings such as wa eguide a ays o Bose-Eins ein condensa es in op ical la ices)
disc e e nonlinea Sch ¨odinge (DNLS) [34] se ing? A pa ial esul has s emmed om he s a is ical analysis o [35], which
concluded ha he e is a la ge p opensi y owa ds eak wa es nea he in eg able limi .
O e he pas ew yea s, we ha e a emp ed o add ess some o he ele an ex ensions o he unde s anding o ogue wa es
pas he s ic ealm o in eg able sys ems. On he one hand, some o he p esen au ho s ha e a emp ed o de elop ogue-wa e
iden i ying me hods ha go beyond he in eg able ealm [36, 37] such as acing hese solu ions as ixed poin s in space- ime. On
he o he hand, a di e en subse o he p esen au ho s has de eloped echniques o unde s anding he s abili y o hese s a es,
conside ing he Floque analysis o he KM wa es and examining he Pe eg ine s a es as na u al limi ing s a es he eo [38].
∗Email: jksulli a[email p o ec ed]
†Email: echa [email protected]
‡Email: jcue [email protected]
§Email: ke ekid@ma h.umass.edu
¶Email: ka [email protected]
2
Dynamical s udies o he e olu ion o gene ic ini ial da a (including expe imen al ones) [39–41] a e ongoing and such examples
exis also in he case o disc e e sys ems [42].
The pu pose o he p esen wo k is mul i- old. We aim o use he so-called Sale no model [43] as a ehicle o going
con ollably beyond he in eg able AL limi and owa ds he physical ealm o he DNLS limi . In p inciple, his is possible since
he model i sel in ol es a homo opy pa ame e , iden i ied as ghe ea e (see, also Eq. (1)) which allows such a (con inuous)
de o ma ion. The hope is ha one can ake his pa h “all he way”, a i ing a physically ele an solu ions o he la e limi .
Indeed, we will show ha in special cases an unp eceden ed possibili y exis s o ca y h ough his p og am. Howe e , gene ically
ou s udy will show ha solu ions in he o m o KM soli ons will exis ; we seek speci ically he la e because no only can we
ob ain he Pe eg ines as a special limi he eo , bu also he s abili y is compu a ionally ac able. We ca y ou he con inua ion
o such s a es and ind su p isingly con olu ed bi u ca ion diag ams. We also obse e ha hese bi u ca ion diag ams (o e g)
can be qui e di e en o di e en b ea he equencies and in special cases may e en con inue o he DNLS limi . Impo an ly
also, we ind ha he looping bi u ca ion diag ams iden i ied come back o “in e sec ” a a di e en poin he in eg able limi ,
p esumably gi ing ise o p e iously unknown, o he bes o ou unde s anding, in eg able sys em solu ions. Hence, ou wo k
may be o in e es bo h o he applica ions o he sys em (bea ing he i s example o such s a es “ eaching” he DNLS limi ),
bu also o he ma hema ical analysis o he in eg able sys em (p o iding p e iously unknown solu ions wi h oscilla o y ails in
he AL limi ).
Ou p esen a ion is s uc u ed as ollows. In he nex sec ion, we o e he gene al amewo k o he model and he analy ical
esul s om he s abili y analysis o he plane wa e solu ions. Then in Sec ion III, we examine ou nume ical indings o
di e en equencies o he pe iodic s a e ωand o e he Sale no-model homo opic con inua ion s eng h g. Finally, in Sec ion
IV we summa ize ou indings and o e some possible di ec ions o u u e s udy.
II. MODEL AND THEORETICAL SETUP
A. Model and b ea he solu ions
The model ha his wo k ocuses on is he so-called Sale no model [43] gi en by
i˙
Ψn+C(Ψn+1 −2Ψn+ Ψn−1) + g(Ψn+1 + Ψn−1)|Ψn|2+ 2 (1 −g)|Ψn|2Ψn= 0,(1)
whe e Ψn:= Ψn( ) : Z×R→Cis he complex wa e unc ion o he n h la ice si e (n∈Z) and o e do deno es di e en ia ion
wi h espec o ime. The pa ame e g∈[0,1] is he homo opy pa ame e whence he DNLS [34] and AL [26, 27] models a e
ob ained om Eq. (1) o alues o g= 0 and g= 1, espec i ely. I hs ands o he dis ance be ween adjacen nodes, hen
C= 1/h2in Eq. (1). Recall ha he model is used due o i s e sa ili y. On he one hand, we can bene i om he knowledge o
analy ical solu ions o he Pe eg ine soli on o he KM wa e o m a he AL limi , ye we can also a emp o connec ( h ough
con inua ions in g) hese indings o he physically applicable (non-in eg able) DNLS limi . Upon inse ing he sepa a ion o
a iables ansa z
Ψn=ψne2iq2 (2)
in o Eq. (1), we a i e a
i˙
ψn+C(ψn+1 −2ψn+ψn−1) + g(ψn+1 +ψn−1)|ψn|2+ 2 (1 −g)|ψn|2−q2ψn= 0,(3)
whe e he pa ame e q ixes he backg ound ampli ude.
A he AL limi , i.e., when g= 1 and C= 1, Eq. (3) possesses a ime-pe iodic solu ion, he Kuzne so -Ma b ea he [32]
gi en explici ly by
ψn( ) = qcos (ω +iθ) + Gcosh ( n)
cos (ω ) + Gcosh ( n),(4)
wi h equency ω( ela ed o he pe iod o he solu ion ia T= 2π/ω), θ=−a csinh (ω), = a ccosh ([2 + cosh (θ)] /3) and
G=−ω/ √3 sinh ( ).
Away om he in eg able AL limi , explici analy ical solu ions a e no longe a ailable. Thus, we iden i y ime-pe iodic
solu ions o pe iod Tby conside ing a empo al disc e iza ion in e ms o he Fou ie se ies expansion:
ψn( ) =
∞
X
m=−∞
φn,meimω ,(5)
3
whe e φn,m a e he Fou ie coe icien s. Upon plugging Eq. (5) in o Eq. (3), we a i e a a se o algeb aic equa ions:
−(mω + 2q2)φn,m +C(φn+1,m −2φn,m +φn−1,m)−2q2
+gX
m′X
m′′ {φn,m′φ∗
n,m′′ (φn−1,m−m′+m′′ +φn+1,m−m′+m′′ )
+2(1 −g)φn,m′φ∗
n,m′′ φn−1,m−m′+m′′ }= 0.
(6)
This sys em is sol ed by a ixed-poin i e a ion me hod, e.g., he New on-Raphson me hod. To his aim, one mus i s ly unca e
he in ini e spa ial la ice in o a ini e one o ende he sys em ac able. He e we choose n∈[−N/2, N/2],N∈2Z+
supplemen ed wi h pe iodic bounda y condi ions ψ−N/2( ) = ψN/2( ),∀ . This way, he o al numbe o nodes is N+ 1.
On he o he hand, one mus unca e he in ini e ( empo al) Fou ie se ies o Eq. (5). We use |m| ≤ 41 in ou compu a ions
p esen ed below. Fo gi en ω,Cand g, we iden i y ime-pe iodic solu ions wi h high accu acy by imposing a s ic ole ance
c i e ion on he no m o successi e i e a es which is wi hin 10−11. I should be no ed in passing ha upon con e gence o he
New on-Raphson me hod, we cons uc he ime-pe iodic solu ion by means o Eq. (5) [38]. The esul ing, by cons uc ion, ime-
pe iodic solu ion can be used as an ini ial condi ion (a = 0) o a ime-s epping scheme o examine i s dynamical e olu ion.
Addi ionally, i can also be used in a Floque analysis-based s abili y compu a ion o assess he spec al s abili y o he solu ion
as we now discuss.
B. Modula ional ins abili y and Floque analysis
The modula ional ins abili y (MI) o he asymp o ic s a e (cons an backg ound) Ψn=qas |n| → ∞ o Eq. (1) (see, he
seminal wo ks o [44, 45] as well as he ecen wo k o [46] on he subjec ) is in es iga ed by conside ing he plane wa e
solu ion
Ψn=qei(kn−ω ), k = 2πM √C
N!, M ∈Z,(7)
wi h equency ωand wa enumbe k. I we inse Eq. (7) o Eq. (1) we ob ain he ollowing dispe sion ela ion:
ω= 4 C+gq2sin2k
2−2q2.(8)
We can now explo e he linea s abili y o he plane wa e solu ion o Eq. (7) by in oducing he ansa z
e
Ψn=hq+ε(a+ib)ei(Qn−ω )iei(kn−ω ), ε ≪1, Q = 2πM′ √C
N!, M′∈Z,(9)
whe e bo h aand ba e ime-dependen , eal- alued unc ions, and Qand Ωco espond o he wa enumbe and equency o he
pe u ba ion, espec i ely. Upon plugging Eq. (9) in o Eq. (1), we ob ain a o de O(ε) he MI dispe sion ela ion gi en by
Ω2= 2 gq2+C[cos (Q+k)−cos (k)] 2gq2+Ccos (Q+k) + 2 gq2−Ccos (k) + 4 (1 −g)q2.(10)
I his condi ion is sa is ied, yielding eal equencies Ω o a gi en pe u ba ion wa enumbe Q, hen he ele an wa enumbe
is s able. On he o he hand, he exis ence o Q’s associa ed wi h complex Ωleads o dynamical ins abili y o he backg ound.
The s abili y o ime-pe iodic solu ions wi h pe iod Tiden i ied ia ixed-poin i e a ions (see, Sec. III) and deno ed as ψ0
nis
examined by conside ing he pe u ba ion ansa z
e
ψn=ψ0
n+ε ξn( ), ε ≪1,(11)
whe e ξn∈Cis he pe u ba ion imposed a he n h si e o he la ice. Then, we inse Eq. (11) in o Eq. (3) and ob ain a o de
O(ε) he go e ning equa ion o he pe u ba ion ξn:
i˙
ξn=−C(ξn+1 −2ξn+ξn−1)−2gψ0
n+1 +ψ0
n−1Re ξn(ψ0
n)∗−g(ξn+1 +ξn−1)|ψ0
n|2
−2 (1 −g)2|ψ0
n|2ξn+ξ∗
n(ψ0
n)2+ 2q2ξn.(12)
Then, he eigen alues λo he so-called monod omy ma ix Ms emming om:
Re(ξn(T))
Im(ξn(T)) =MRe(ξn(0))
Im(ξn(0)) .(13)
4
-1 0 1 2
-1
-0.5
0
0.5
1
5 10 15 20
0
0.2
0.4
0.6
0.8
1
FIG. 1: (Colo online) The le panel p esen s he spec um o a(n uns able) ime-pe iodic solu ion wi h ω= 8 ( illed black ci cles). No e ha
he heo e ically p edic ed uns able modes ia Eq. (10) a e shown wi h ed open ci cles he ein. The igh panel demons a es a compa ison
be ween he Floque exponen s Re(iΩ) o ime-pe iodic solu ions (solid black lines) and he ones o he MI analysis (dashed-do ed ed lines)
o Eq. (10) o e ω. No e ha bo h panels co espond o he case wi h g= 1 (as well as C= 1 and q= 1/√2).
de e mine he s abili y ai o o a ime-pe iodic solu ion ψ0
n. Those eigen alues a e he so-called Floque mul iplie s. In
pa icula , as he sys em is symplec ic and Hamil onian, a solu ion is deemed neu ally s able i all he Floque mul iplie s
λ=λ +iλio Mlie on he uni ci cle. I |λ|>1, hen wo ypes o ins abili ies can a ise. I he mul iplie s a ise in eal
pai s ha a e away om he uni ci cle, he ins abili y is conside ed exponen ial due o he exponen ial g ow h o he associa ed
pe u ba ions. On he con a y, i he mul iplie s a ise in complex qua e s (inside and ou side he uni ci cle), hen he ins abili y
is deemed oscilla o y. We conclude his sec ion by men ioning in passing ha he eigen equency o he pe u ba ion Ωconnec s
wi h he Floque mul iplie s λ ia
λ=eiΩT.(14)
III. NUMERICAL RESULTS
The a ailabili y o he exac solu ion (4) o he case wi h g= 1, i.e., he AL limi allows us o no only benchma k ou
nume ical me hods bu also o compu e he Floque mul iplie s di ec ly om Eq. (13). He ea e , we conside a la ice wi h
N= 100 si es and se C= 1 and q= 1/√2in Eqs. (6) and (12) (all o ou nume ical esul s discussed in his sec ion we e
ob ained o hese choices). A i s , we compu e he Floque mul iplie s om Eq. (13) ( o g= 1) by using he ini ial- alue-
p oblem (IVP) sol e DOP853 [47] wi h ( ela i e and absolu e) ole ances 10−11. Al hough a non-s i and explici IVP sol e ,
DOP853 can pe o m a ime s ep-size adap a ion ( o sa is y he use -speci ied ole ance c i e ia) o s i egions by educing he
ime s ep-size. Indica i ely, Fig. 1 summa izes ou indings on he Floque mul iplie s o he exac solu ion (4). The le panel o
he igu e p esen s he Floque mul iplie s o he exac ime-pe iodic solu ion wi h ω= 8 ( illed black ci cles) oge he wi h he
heo e ically p edic ed uns able modes om Eq. (10) ia Eq. (14) (open ed ci cles). As a as he KM b ea he is conce ned, he
p esence o Floque mul iplie s wi h |λ|>1 ende he solu ion uns able. In addi ion, i is e iden om his panel ha a subse
o he uns able modes o he KM b ea he coincides wi h hose o he modula ionally uns able backg ound. Howe e , he e exis
o he uns able eigenmodes ha de ia e om he la e due o he p esence o he localized solu ion pe u bing he modes o he
backg ound. This is na u al o expec gi en he modula ional ins abili y o he backg ound (analyzed in he p e ious sec ion)
on op o which he pe iodic solu ion “li es”. The igh panel o Fig. 1 complemen s ou analysis o he exac KM b ea he
and demons a es he dependence o i s uns able Floque exponen s Re(iΩ) on ω(solid black line) oge he wi h he espec i e
esul s o he MI analysis (dashed-do ed ed line). I can be disce ned om his panel ha he Floque exponen s o he KM
b ea he app oach he asymp o ic alues heo e ically p edic ed by Eq. (10).
We now wish o explo e he exis ence and s abili y o ime-pe iodic solu ions o e he ele an pa ame e space by going
beyond he well-de ined and analy ically ac able case o he in eg able limi o g= 1. We hus a y he pa ame e g(which is
used as ou bi u ca ion pa ame e ) by employing a pseudo-a cleng h con inua ion algo i hm [48, 49], and conside di e en ixed
alues o he pe iod T. I should be no ed ha he pseudo-a cleng h con inua ion algo i hm is capable o passing h ough u ning
poin s (whe e he Jacobian o he sys em o equa ions is singula and hus non-in e ible) and acing (connec ed) b anches o
solu ions. The nume ical esul s epo ed in his wo k we e ob ained by using a ixed and ela i ely small a cleng h s ep-size
o 10−2in o de o p e en he p edic o -co ec o s ep om con e ging o a solu ion o a nea by b anch. As pe he di ec
nume ical simula ions epo ed below, we use again he DOP853 [47] me hod o ad ancing Eq. (1) o wa d in ime (and wi h
he same ole ance c i e ia as be o e).
Figu e 2 summa izes ou esul s on he exis ence o o ime-pe iodic solu ions o he Sale no model [c . Eq. (1)]. In pa icula ,
we demons a e he dependence o max (|ψ|)on g o ω= 5,ω= 25, and ω= 10 in he le , middle, and igh panels,
5
0.98 0.99 1 1.01
6
8
10
12
0.99 1 1.01
15
20
25
30
35
0 0.2 0.4 0.6 0.8 1
5
10
15
0.9 0.95 1 1.05
10
12
14
16
FIG. 2: (Colo online) The l∞-no m o |ψn|as a unc ion o gco esponding o ime-pe iodic solu ions o alues o ω= 5,ω= 25 and
ω= 10, espec i ely. The labels (a)-(e) in he le and igh panels a e associa ed wi h Figs. 3 and 4 ha ollow.
espec i ely. No e ha he labels (a)-(e) in he le and igh panels a e connec ed o Figs. 3 and 4, espec i ely. Based on he
panels o Fig. 2, a cascade o u ning poin s is clea ly e iden al hough all o hem sugges an in iguing inding ha we discuss
now. We conside i s he le panel o he igu e co esponding o ω= 5. S a ing om g= 1, he ime-pe iodic solu ion
depa s om he in eg able limi (i.e., he AL limi ), heading o smalle alues o gun il i eaches a u ning poin a g≈0.981
upon which i comes back o g= 1, i.e., he AL limi , and hen ollows a “snake” pa e n wi h se e al c ossings o he AL limi .
I should be no ed ha we s opped ou con inua ion algo i hm a g= 1 ( he e minal solu ion is labeled by (e) he ein). Based
on ou addi ional nume ical in es iga ions, his is en ou e o mul iple addi ional c ossings o g= 1 ( esul s a e no shown). The
labels o he le panel a e connec ed wi h he p o o ypical con igu a ions o his case in Fig. 3. In pa icula , i s le column
p esen s he spa ial dis ibu ion o he densi ies o he ime-pe iodic solu ions, i.e., |ψn|2 o alues o g= 1 (panels (a), (c)-(e)),
and g= 0.982 (panel (b)), espec i ely. The inse s he ein co espond o he associa ed Floque mul iplie s which hemsel es
sugges ha all compu ed solu ions a e highly uns able wi h a dominan uns able mode o he o de o λ ∼ O(1). Again, his
can be na u ally expec ed on he basis o he ins abili y o he backg ound. The s iking ea u e o he solu ions p esen ed in
panels (b)-(e) is he o ma ion o an oscilla o y backg ound (o small ampli ude) in con as wi h he KM b ea he o panel (a)
whe e he localized wa e si s a op a cons an backg ound. Such p o iles ea u ing small in-ampli ude wa e ains a e s ongly
eminiscen o nanop e a, and o he bes o ou knowledge a e i s epo ed o he Sale no model in he p esen wo k (see,
also he ecen wo ks o [50, 51] o Toda la ices and he DNLS wi h sa u a ion). The o he s iking ea u e o ou indings
is ha he KM b ea he (4) o he AL model (g= 1) seems no o be he only solu ion a ha limi , as his has al eady been
e iden in panels (b)-(e) o Fig. 3. Al hough he in es iga ion o hose ex a solu ions by using in eg able sys ems echniques is
o undamen al impo ance, i is beyond he scope o ou p esen wo k. We summa ize ou esul s in his case by p esen ing he
espec i e spa io- empo al dynamics in he igh panel o Fig. 3. In pa icula , con ou plo s o he spa io- empo al e olu ion o
he densi y o ime-pe iodic solu ions a e shown o one pe iod (T= 2π/5in his case). Though he solu ions emain obus o
one pe iod, hus alida ing ou nume ical app oach o iden i ying hem ( ia ixed-poin i e a ions), we obse ed he eme gence
o he ins abili y which happens a la e imes ( esul s no shown) leading o a b eakdown o he localized wa e o m.
Simila ly, he middle panel o Fig. 2 co esponds o he case o ω= 25. I p esen s a ype o spi al s uc u e a ound he KM
solu ion ha e en ually leads o a p og essi ely mo e dense loop s uc u e ha keeps epea ing i sel in he cou se o he pseudo-
a cleng h con inua ion. See also he inse he ein which sugges s ha complex s uc u es may a ise also a a ine scale wi hin
he bi u ca ion diag am. The iden i ied pe iodic s a es bea simila ea u es as be o e, including p og essi ely mo e p onounced
(along he ele an b anches) ails in he ele an wa e o ms. Fu he mo e, in line wi h ea lie (MI-based and compu a ional)
indings and gene al expec a ions, he solu ions a e gene ically ound o be uns able, due o he ins abili y o hei espec i e
backg ound.
I is pe haps impo an o highligh wo pe ading ea u es ega ding he na u e o ou esul s (bo h he ones abo e, as well
as he ones ha will ollow). The i s one is ha o he bes o ou knowledge he esul s abo e (and below) cons i u e he i s
de ini i e iden i ica ion o ogue-like pa e ns in disc e e (nonlinea dynamical la ice) sys ems beyond he ex emely impo an ,
ye p ac ically limi ed ealm o in eg able sys ems. Secondly, ou indings a e no solely o in e es o non-in eg able dispe si e
sys em p ac i ione s, bu hey a e also a mo i a ion o u he in eg able sys em in es iga ions. The panel (b) o Fig. 3, o
example, sugges s he exis ence o KM- ype solu ions on op o a s a iona y nanop e onic backg ound which is ime-independen
in he spi i o ecen wo ks o [52, 53] in co esponding con inuum limi p oblems. These a e s a es ha we belie e a e eminen ly
ele an o explo e in an analy ical o m wi hin he amewo k o in eg able sys ems (al hough his is ou side he scope o he
p esen s udy).
We conclude his sec ion o ou indings by going h ough he igh panel o Fig. 2 and he associa ed esul s shown in Fig. 4
co esponding o he case o ω= 25. Based on he o me , we obse e again a cascade o c ossings happening a he AL limi
(see, he inse he ein), howe e he e he bi u ca ion cu e s a s heading owa ds smalle alues o gun il i eaches g= 0,

6
-20 -10 0 10 20
0
20
40
60 (a)
0
-1 0 1
-1
0
1
0123
-1
0
1
-20 -10 0 10 20
0
0.5
1
0
10
20
30
40
50
-20 -10 0 10 20
0
10
20
30
40
50 (b)
-1 0 1
-1
0
1
-10 -5 0
-1
0
1
0
0.3
0.6
0.9
-20 -10 0 10 20
0
0.5
1
0
10
20
30
40
-20 -10 0 10 20
0
10
20
30
40
50 (c)
-1 0 1
-1
0
1
-1 0 1 2 3
-1
0
1
0
0.3
0.6
0.9
-20 -10 0 10 20
0
0.5
1
0
10
20
30
40
-20 -10 0 10 20
0
30
60
90 (d)
-1 0 1
-1
0
1
-1 0 1 2 3
-1
0
1
0
1
2
-20 -10 0 10 20
0
0.5
1
0
20
40
60
80
-20 -10 0 10 20
0
40
80
120 (e)
-1 0 1
-1
0
1
-1 0 1 2 3
-1
0
1
-30 -25 -20 -15
0
0.3
0.6
0.9
-20 -10 0 10 20
0
0.5
1
0
40
80
120
FIG. 3: (Colo online) Nume ical esul s on ime-pe iodic solu ions o he Sale no model (3) o ω= 5 associa ed wi h he bi u ca ion cu e
o he le panel o Fig. 2 (see, he labels he ein). The le column co esponds o he spa ial dis ibu ion o |ψn|2 o alues o (a), (c)-(e)
g= 1 and (b) g= 0.982, espec i ely. The inse s shown on he le and igh o each panel p o ide close-ups o he p o iles as well as Floque
spec a. The igh column p esen s he spa io- empo al e olu ion o |ψn|2o he p o iles shown in he le column o one pe iod (T= 2π/5).
7
-20 -10 0 10 20
0
50
100
150 (a)
-1 0 1 2
-1
0
1
-20 -10 0 10 20
0
0.2
0.4
0.6 0
50
100
150
-20 -10 0 10 20
0
20
40
60
80 (b)
-40-30-20-10
0
0.3
0.6
0.9
-1 0 1
-1
0
1
-20 -10 0 10 20
0
0.2
0.4
0.6 0
20
40
60
80
-20 -10 0 10 20
0
90
180
270 (c)
-1 0 1
-1
0
1
-50 -25 0
0
0.4
0.8
1.2
-20 -10 0 10 20
0
0.2
0.4
0.6 0
90
180
270
-20 -10 0 10 20
0
20
40
60
80 (d)
-1 0 1
-1
0
1
-50-35-20 -5
0
0.4
0.8
1.2
-20 -10 0 10 20
0
0.2
0.4
0.6 0
35
70
-20 -10 0 10 20
0
2
4
6
8(e)
-1 0 1 2
-1
0
1
-20 -10 0 10 20
0
2
4
60
2
4
6
FIG. 4: (Colo online) Same as Fig. 3 bu o he case o ω= 10 associa ed wi h he igh panel o Fig. 2. Again, he le and igh columns
co espond o he spa ial dis ibu ion o |ψn|2and i s spa io- empo al e olu ion, espec i ely. The in eg a ion was pe o med o e one pe iod
o (a)-(d) and o e 10 pe iods o panel (e), espec i ely. The inse s shown on he le and igh o each panel p o ide close-ups o he p o iles
as well as Floque spec a. Panels (a)-(c) co espond o g= 1 whe eas he ones o (d) and (e) o g= 0.9and g= 0 (DNLS case), espec i ely.
8
i.e., he DNLS limi . No ice how his appea s o be an in e media e case be ween he less o de ed and mo e expanded diag ams
o small ωand he mo e o de ed and con ined diag am o ω= 25. The p o iles shown in panels (a)-(c) in Fig. 4 co espond
o alues o g= 1 whe eas he ones o (d) and (e) o g= 0.9and g= 0, espec i ely. Again, a common inding is ha all
ime-pe iodic solu ions iden i ied in his wo k ea u e an oscilla o y backg ound as soon as we depa om he in eg able limi
(see, he le inse s in he le column o Fig. 4) as well as hey a e uns able (see, he igh inse s he ein). This e ec is e en
mo e p onounced o he case wi h g= 0 as is shown in panel (e) o he igu e. To he bes o ou knowledge, such ime-pe iodic
solu ions o DNLS on a backg ound ha e no appea ed in he li e a u e so a ; indeed we a e only awa e o such e ec i ely
quasi-pe iodic solu ions o he model on op o a anishing backg ound pe he wo k o [54]. This solu ion is also uns able as is
e iden in he Floque spec um shown in he inse o he panel, al hough i emains obus o e many pe iods o ime in eg a ion.
Indeed, he igh panel he ein demons a es he spa io- empo al e olu ion o he densi y again o e 10 pe iods whe e |ψ0( )|2
b ea hes o e ime. I should also be no ed ha he oscilla o y backg ound is a he isible in his case (as well as he one o
panels (b) and (d)) bu again i emains s eady o e he ime e olu ion, i.e., co esponding o a de ini i e nanop e onic s a e in
his se ing.
IV. CONCLUSIONS AND FUTURE CHALLENGES
In his wo k, we made an a emp o explo e he exis ence, s abili y and dynamics o ime-pe iodic solu ions o he Sale no
model. This was wi h a h ee- old scope in mind: i s ly, o es ablish ha ele an solu ions such as he KM a e no unique o
pa icula o he in eg able limi , bu can be con inued o gene ic non-in eg able alues o he homo opic pa ame e g. Secondly,
we wished o explo e whe he addi ional in iguing solu ions could a ise in he in eg able model, a ea u e ha was b ough o h
om ou pseudo-a cleng h con inua ion esul s o pe iodic o bi s. Las ly, we in ended o examine whe he some o he ele an
solu ions could be con inued o he DNLS limi o g= 0; he e, we ound ha o sui able choice o he b ea he equency
indeed ha was also possible. Upon employing ixed-poin me hods, we iden i ied he pe inen wa e o ms using New on’s
me hod ( o pe iodic o bi s) and hei s abili y was in e ed by pe o ming a Floque analysis. The use o pseudo-a cleng h
con inua ion allowed us o pe o m a pa ame ic con inua ion o e he homo opy pa ame e gand his p o ed o be c ucial in
un a eling he complexi y o he possible solu ions in he Sale no model. Addi ionally o he in eg able limi o g= 1 we
iden i ied mul iple ime-pe iodic solu ions ha si a op o an oscilla o y backg ound being s ongly eminiscen o nanop e a.
Ano he s iking inding o ou wo k was he ime-pe iodic solu ion which was iden i ied a he DNLS limi , i.e., g= 0. To he
bes o ou knowledge, such a wa e o m o he DNLS (on op o a non- anishing backg ound) was no p e iously epo ed.
Based on he abo e indings and compu a ional echniques ha we ha e de eloped in his wo k, he e a e clea ly many
di ec ions o u u e s udies. A he AL limi , a po en ial analysis o he Zhaka o -Shaba p oblem o he nanop e onic solu ions
epo ed in Sec ion III will be o pa amoun impo ance in o de o de i e hem in possibly closed o m (analogously o hei
iden i ica ion in con inuum p oblems [52, 53]). On he o he hand, i is wo h in es iga ing he g= 0 solu ion epo ed in he
p esen wo k, and in pa icula o s udy he con igu a ion space o solu ions as a unc ion o ωin his case. This will pa e he
way owa ds po en ially iden i ying Pe eg ine-like en i ies o he DNLS as ime-pe iodic solu ions a he limi T→ ∞(ω≫1).
Ano he impo an pa h o explo e is he con inua ion o he p esen ime-pe iodic solu ions as he dis ance be ween adjacen si es
dec eases, hus app oaching he con inuum limi when C= 1/h2inc eases. This way, we will be able o connec ou indings
wi h he esul s o he ( ocusing) NLS. A subse o he abo e compu a ional s udies can be ca ied ou qui e e icien ly wi h
he use o s a e-o - he-a bi u ca ion packages such as AUTO [55] and COCO [56] which addi ionally allow b anch swi ching,
among o he ea u es. Such di ec ions a e p esen ly unde conside a ion and will be epo ed in u u e publica ions.
Acknowledgmen s
J.C.-M. was suppo ed by MAT2016- 79866-R p ojec (AEI/FEDER,UE). PGK acknowledges suppo om he U.S. Na ional
Science Founda ion unde G an s no. PHY-1602994 and DMS-1809074 (PGK). He also also acknowledges suppo om he
Le e hulme T us ia a Visi ing Fellowship and he Ma hema ical Ins i u e o he Uni e si y o Ox o d o i s hospi ali y du ing
pa o his wo k.
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