Wa e Mo ion 137 (2025) 103547
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0165-2125/© 2025 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license
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On he p oximi y o Ablowi z–Ladik and disc e e nonlinea
Sch ödinge models: A heo e ical and nume ical s udy o
Kuzne so -Ma solu ions
Madison L. Ly le a,b, E s a hios G. Cha alampidis a,c,∗, Dionyssios Man za inos d,
Jesus Cue as-Ma a e e, , Panayo is G. Ke ekidis g, Nikos I. Ka achalios h
aMa hema ics Depa men , Cali o nia Poly echnic S a e Uni e si y, San Luis Obispo, 93407-0403, CA, USA
bDepa men o Applied Ma hema ics and S a is ics, Colo ado School o Mines, Golden, 80401, CO, USA
cDepa men o Ma hema ics and S a is ics, and Compu a ional Science Resea ch Cen e , San Diego S a e Uni e si y, San
Diego, 92182-7720, CA, USA
dDepa men o Ma hema ics, Uni e si y o Kansas, Law ence, 66045, KS, USA
eG upo de Física No Lineal, Depa amen o de Física Aplicada I, Uni e sidad de Se illa. Escuela Poli écnica Supe io , C/ Vi gen de Á ica,
7, 41011 Se illa, Spain
Ins i u o de Ma emá 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
gDepa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s Amhe s , Amhe s , 01003-4515, MA, USA
hDepa men o Ma hema ics, Uni e si y o Thessaly, Lamia, 35100, G eece
A R T I C L E I N F O
Keywo ds:
B ea he s
Rogue wa es
In eg able and non-in eg able la ice sys ems
Bi u ca ion analysis
A B S T R A C T
In his wo k, we in es iga e he o ma ion o ime-pe iodic solu ions wi h a non-ze o backg ound
ha emula e ogue wa es, known as Kuzne so -Ma (KM) b ea he s, in physically ele an
la ice nonlinea dynamical sys ems. S a ing om he comple ely in eg able Ablowi z–Ladik
(AL) model, we demons a e ha he e olu ion o KM ini ial da a is p oximal o ha o he
non-in eg able disc e e Nonlinea Sch ödinge (DNLS) equa ion o ce ain pa ame e alues o
he backg ound ampli ude and b ea he equency. This inding p omp s us o in es iga e he
dis ance (in ce ain no ms) be ween he e ol ed solu ions o bo h models, o which we igo -
ously de i e and nume ically con i m an uppe bound. Finally, ou s udies a e complemen ed
by a wo-pa ame e (backg ound ampli ude and equency) bi u ca ion analysis o nume ically
exac , KM- ype b ea he solu ions o he DNLS equa ion. Alongside he s abili y analysis o
hese wa e o ms epo ed he ein, his wo k addi ionally showcases po en ial pa ame e egimes
whe e such wa e o ms wi h a la backg ound may eme ge in he DNLS se ing.
1. In oduc ion and mo i a ion
The s udy o dispe si e nonlinea la ice dynamical models has been a opic o conside able in e es o e he pas decades [1,2].
Among he physical a eas mo i a ing he ele an de elopmen s, one can single ou he pa icula con ibu ions om he s udy
o op ical wa eguides [3] (bu also con inuum pho o e ac i e media wi h pe iodic po en ials) and he explo a ion o mean- ield
a omic Bose–Eins ein condensa es (BECs) in he p esence o pe iodic ex e nal (so-called op ical la ice) po en ials [4]. Howe e ,
∗Co esponding au ho a : Depa men o Ma hema ics and S a is ics, and Compu a ional Science Resea ch Cen e , San Diego S a e Uni e si y, San Diego,
92182-7720, CA, USA.
E-mail add esses: [email p o ec ed] (M.L. Ly le), [email p o ec ed] (E.G. Cha alampidis), [email p o ec ed] (D. Man za inos),
[email p o ec ed] (J. Cue as-Ma a e ), [email p o ec ed] (P.G. Ke ekidis), [email p o ec ed] (N.I. Ka achalios).
h ps://doi.o g/10.1016/j.wa emo i.2025.103547
Recei ed 20 Decembe 2024; Recei ed in e ised o m 12 Ma ch 2025; Accep ed 20 Ma ch 2025
Wa e Mo ion 137 (2025) 103547
2
M.L. Ly le e al.
ele an models, compu a ions, and expe imen s a e by no means limi ed o hese sub ields; a he , hey b oadly ex end o o he
con ex s, including, among o he s, nonlinea a ian s o elec ical ci cui s [5], elas ically in e ac ing beads wi hin g anula c ys al
me ama e ials [6–8], supe conduc ing Josephson junc ion a ays [9,10], mic omechanical a ays o can ile e s [11], and DNA
dena u a ion models [12].
The e has been a conside able weal h o models ele an o hese spa ially disc e e applica ions, including Klein–Go don and
Fe mi–Pas a–Ulam–Tsingou [13,14]. Howe e , he mos uni e sal dispe si e la ice nonlinea model is, a guably, he disc e e
nonlinea Sch ödinge (DNLS) equa ion [15,16]. I se es as he p o o ypical ehicle o he s udy o soli a y wa es, ins abili ies, and
dynamics in disc e e nonlinea op ics [17] while also holding conside able ele ance in a omic BECs [4]. To his day, i con inues o
play a subs an ial ole in mode n de elopmen s conce ning, e.g., opological la ices [17], la bands [18,19], and many o he s. F om
a ma hema ical physics pe spec i e, i has an addi ional, pa icula ly in e es ing ea u e: he exis ence o an in eg able analog [20].
This, in u n, c ea es he po en ial o nume ous s udies conce ning he b eaking o in eg abili y, pe u ba ion e ec s on conse a ion
laws, soli a y ea u es, e c. in sui able in e pola ions be ween he in eg able and non-in eg able a ian s o he model [21].
An aspec o DNLS soli a y wa es ha has been o in e es in ecen yea s is i s po en ial o ea u e la ge-ampli ude ogue o
eak wa es. The s udy o such wa es [22] has ecen ly a ac ed conside able a en ion due o he eme gence o expe imen al
capabili ies ha enable he de ec ion and isualiza ion o hese pa e ns a he le el o nonlinea wa es in op ical sys ems [23–26],
luid se ings in wa e anks [27–29], plasmas [30], and e en BECs [31]. These de elopmen s ha e been summa ized in a wide
ange o e iews, including [32–36]. In he disc e e ealm, he wo k o [37] showcased he exis ence o all he cen al nonlinea
wa e o ms in he in eg able Ablowi z–Ladik (AL) model: i.e., he Pe eg ine soli on (P) [38], he Akhmedie b ea he (AB) [39], as
well as he Kuzne so -Ma (KM) b ea he [40,41].
The eme gence o such pa e ns in disc e e in eg able media aised he ques ion o hei po en ial obse abili y [42,43]. When
conside ing models in e pola ing be ween he in eg able (AL) and non-in eg able (DNLS) limi , hese wo ks se he expec a ion ha
ex eme e en s a e mo e likely o exis in he o me , a he han in he la e . On he o he hand, in ecen yea s, some o he
p esen au ho s ha e explo ed he p oximi y be ween he DNLS and he AL model h ough analy ical es ima es [44,45] as well
as con inua ions be ween he wo ia nume ical compu a ions [46]. The p esen wo k aims o b ing hese di e en elemen s o
he li e a u e oge he , ope a ing a he nexus o ex eme solu ions o la ice nonlinea dynamical sys ems o bo h he in eg able
(AL) and he non-in eg able (DNLS) kind, and explo ing he po en ial con inua ion o KM solu ions om he o me owa ds he
la e . Mo eo e , he p esen wo k conside s such ques ions om he complemen a y pe spec i es o igo ous es ima es, as well
as o nume ical compu a ions in ol ing he exis ence, spec al s abili y, and nonlinea dynamics o such s a es. We ind ha he
ele an s a es can o en be con inued and we can iden i y co esponding wa e o ms in he DNLS limi . This is co obo a ed by he
p esence o he analy ical es ima es. I is en isioned ha he indings he ein may mo i a e op ical wa eguide expe imen s ha may
enable he iden i ica ion o he KM solu ions p esen ed he ein. Such expe imen s should be di ec ly accessible in a ays o op ical
wa eguides [3], a opic o con inuing in e es in ecen in es iga ions [47]. They may also a ise in an e ec i e o m in Bose–Eins ein
condensa es in op ical la ices, as explained, e.g., in [48]. O pa icula in e es in ha ega d will be he e olu ion o he ele an
s a es and, mo e speci ically, he mani es a ion o hei dynamical ins abili y, in connec ion wi h he esul s p esen ed below.
Ou p esen a ion is s uc u ed as ollows. In Sec ion 2, we p esen he model se up and ele an solu ions o in e es . In Sec ion 3,
we analyze some p elimina y nume ical compu a ions showcasing he p oximi y o e long imes ( o sui able solu ion pa ame e s)
o he KM wa e o ms o he AL and he DNLS models. This inding hen mo i a es Sec ion 4, whe e we p o ide a igo ous analysis
o he g ow h o e ime o he dis ance be ween he solu ions o he DNLS and he AL sys ems in he case o non-ze o bounda y
condi ions. We es ablish ha his g ow h is linea in ime wi h a p e ac o whose dependence on he size o ini ial da a is analyzed.
In Sec ion 5 we p esen a con inua ion o he KM b anch o solu ions o he DNLS model, along wi h a discussion o he spec al
s abili y o he pe inen solu ions. The wo k is summa ized and ou conclusions a e p esen ed in Sec ion 6.
2. The model se up
In ou subsequen heo e ical and compu a ional analysis, we conside he Sale no model [21], explici ly gi en by
i
𝛹𝑛+𝐶(𝛥𝑑𝛹)𝑛+𝑔(𝛹𝑛+1 +𝛹𝑛−1)|𝛹𝑛|2+ 2 (1 − 𝑔)|𝛹𝑛|2𝛹𝑛= 0,(1)
ha in e pola es be ween he comple ely in eg able Ablowi z–Ladik (AL) model [49,50] (a 𝑔= 1) and he disc e e Nonlinea
Sch ödinge (DNLS) equa ion [15] (a 𝑔= 0). In Eq. (1), 𝛹𝑛=𝛹𝑛(𝑡) ep esen s he complex- alued wa e unc ion a a la ice si e
𝑛∈Z and ime 𝑡∈R, (𝛥𝑑𝛹)𝑛=𝛹𝑛+1 − 2𝛹𝑛+𝛹𝑛−1, and he coupling cons an 𝐶 be ween nea es neighbo ing si es is gi en by
𝐶= 1∕ℎ2 wi h ℎ being he la ice spacing, i.e., he dis ance be ween adjacen si es.
Ou main in e es is he s udy o b ea he solu ions o Eq. (1) on a non-ze o backg ound. Since such solu ions in ol e wo
equencies, i.e., one associa ed wi h he b ea he i sel and he o he wi h backg ound oscilla ions, we conside he sepa a ion o
a iables ansa z:
𝛹𝑛=𝜓𝑛𝑒2𝑖𝑞2𝑡, 𝑞 > 0, 𝜓𝑛∈C,(2)
whe e 𝑞 ep esen s he backg ound ampli ude o he solu ion o in e es , and hus 2𝑞2 is he oscilla ion equency o he
backg ound. Upon inse ing Eq. (2) o Eq. (1), we ob ain:
i𝜓𝑛+𝐶(𝛥𝑑𝜓)𝑛+𝑔(𝜓𝑛+1 +𝜓𝑛−1)|𝜓𝑛|2+ 2 [(1 − 𝑔)|𝜓𝑛|2−𝑞2]𝜓𝑛= 0,(3)
Wa e Mo ion 137 (2025) 103547
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M.L. Ly le e al.
which is he p incipal model equa ion o in e es o s udying ime-pe iodic solu ions 𝜓𝑛(𝑡) = 𝜓𝑛(𝑡+𝑇𝑏) o pe iod 𝑇𝑏= 2𝜋∕𝜔𝑏 and
equency 𝜔𝑏. Fo 𝐶= 1, he AL model (𝑔= 1) admi s explici ogue wa e solu ions [37] including he (disc e e) Kuzne so -Ma
(KM) b ea he , Pe eg ine (P) soli on, and (spa ially-pe iodic) Akhmedie b ea he s (AB). In pa icula , he KM b ea he solu ion o
he AL model is a ime-pe iodic solu ion wi h pe iod 𝑇𝑏(= 2𝜋∕𝜔𝑏), and is explici ly gi en by:
𝜓𝑛(𝑡) = 𝑞cos (𝜔𝑏𝑡+ i𝜃)+𝐺cosh (𝑟𝑛)
cos (𝜔𝑏𝑡)+𝐺cosh (𝑟𝑛),(4)
wi h pa ame e s 𝜃= − a csinh (𝜔𝑏∕(2𝑞2)), 𝑚= (1 + 𝑞2)∕𝑞2, 𝑟= a ccosh [(cosh 𝜃+𝑚− 1)∕𝑚], and 𝐺= −𝜔𝑏∕(2𝑞2√𝑚sinh 𝑟). No e ha
he Pe eg ine b ea he [37,38] is he limi ing case 𝑇𝑏↦∞ (o 𝜔𝑏↦0) o he KM solu ion o Eq. (4). In his wo k, we solely ocus on
s udies e ol ing a ound KM- ype solu ions o bo h he AL (𝑔= 1) and DNLS (𝑔= 0) models. He ea e , we ix 𝐶= 1 co esponding
o a la ice wi h uni spacing, i.e., ℎ= 1.
3. P elimina y nume ical s udies
Recen ly, in [46], he homo opic con inua ion o KM b ea he s (and hei s abili y) in he Sale no model was conside ed, and
nume ical esul s showcased he eme gence o KM- ype b ea he s. Some o hese KM- ype solu ions could be con inued all he way
o he DNLS limi (i.e., 𝑔= 0 in Eq. (1)), bu ea u ed an oscilla o y backg ound. A emp s o iden i ying such solu ions on a la
backg ound by s a ing om he an i-con inuum limi (i.e., 𝐶= 0) a e epo ed in [51], al hough he ele an wa e o ms ob ained
do no enjoy ogue-like beha io . Tha is o say, he wa e o ms he ein do no seem o ‘‘appea ou o nowhe e and disappea
wi hou a ace’’ [52].
He ein, we ake a di e en pa h and examine he possibili y o iden i ying KM solu ions o he DNLS by conside ing he
ecen ad ances on he closeness o localized s uc u es be ween he AL and DNLS models [44,45,53]. In [44], he p oximi y o
Pe eg ine wa e o ms in he AL and DNLS models was examined, showing hei pe sis ence in DNLS models when 𝑞, i.e., he alue
o he backg ound, is small. Mo i a ed by his inding, we now explo e nume ically whe he KM b ea he s wi h small backg ound
ampli udes 𝑞 ≪ 1 and equencies 𝜔𝑏≪1, i.e., close o he Pe eg ine limi , may pe sis in he DNLS model. In ou compu a ions ha
ollow in his sec ion, we u ilize a la ice [−𝑁∕2, 𝑁∕2 − 1] consis ing o 𝑁= 600 nodes (unless s a ed o he wise), whe e pe iodic
bounda y condi ions a e supplemen ed in Eq. (3).
Then, he ini ial- alue p oblem (IVP) consis ing o Eqs. (3)–(4) is sol ed nume ically by using MATLAB’s buil -in Adams–
Bash o h–Moul on ode113 sol e (wi h ela i e and absolu e ole ances 10−13). We checked ou nume ical esul s by addi ionally
using he Buli sch–S oe me hod [54] (wi h same ela i e and absolu e ole ances), and he esul s ma ched exac ly. The ideli y
o ou compu a ions u ilizing bo h nume ical me hods was u he checked by moni o ing he conse ed quan i ies o he AL and
DNLS models, espec i ely gi en by
𝑃AL(𝑡) = ∑
𝑛
log (1 + |𝜓𝑛|2),(5)
and
𝑃DNLS(𝑡) = ∑
𝑛|𝜓𝑛|2.(6)
We ound ha he maximum ela i e e o s, i.e., max (|𝑃AL(𝑡) − 𝑃AL(0)|∕𝑃AL(0)) (and simila ly o he DNLS) we e ∼ 10−14.
Le us now u n ou ocus o he spa io- empo al e olu ion o KM b ea he s [c . Eq. (4)] in Eq. (3) (again wi h 𝐶= 1) o 𝑔= 1
(AL) and 𝑔= 0 (DNLS). We use he analy ical KM solu ion o Eq. (4) wi h 𝜔𝑏= 0.005 and 𝑞= 0.005 as an ini ial condi ion o bo h
models a 𝑡= 0, and ad ance Eq. (3) o wa d in ime o up o 20 pe iods, i.e., 𝑡= 20 × 𝑇𝑏≈ 2.5133 × 104. Ou esul s o hese cases
a e summa ized in Fig. 1. In pa icula , he panel (a) o he igu e depic s he empo al e olu ion o he ampli ude a he 𝑛= 0 si e,
i.e., |𝜓0(𝑡)|, o he AL and DNLS models (i.e., 𝑔= 1 and 𝑔= 0 in Eq. (3)). I can be disce ned om his panel ha he e olu ions o
he ampli udes a e close o a ew pe iods o he ime in eg a ion al hough a phase lag in hei e olu ion g adually ge s mani es ed,
and p og essi ely becomes mo e and mo e p onounced o longe imes. We complemen his i s ound o nume ical simula ions
in panels (b) and (c) o Fig. 1 whe e we summa ize he spa io- empo al e olu ion o he ampli ude |𝜓𝑛(𝑡)| o he solu ions o he
AL and DNLS models, espec i ely. Despi e his phase lag epo ed in panel (a), panels (b) and (c) exhibi simila quali a i e (and
quan i a i e) ea u es in he e olu ion o he wa e o ms.
These indings now beg he ques ion whe he such a KM solu ion can be iden i ied nume ically as a ixed poin , i.e., as a
nume ically exac pe iodic o bi o he DNLS case. To explo e his possibili y, we ollowed he nume ical app oach o [46,51]
in which a 𝑇𝑏= 2𝜋∕𝜔𝑏 ime-pe iodic solu ion is sough by conside ing he Fou ie decomposi ion
𝜓𝑛=
∞
∑
𝑚=−∞
𝑛,𝑚𝑒i𝑚𝜔𝑏𝑡,(7)
whe e 𝑛,𝑚 a e he Fou ie coe icien s and 𝑚 a e he Fou ie modes in ime. Then, upon plugging Eq. (7) in o Eq. (3) we ob ain a
oo - inding p oblem o 𝑛,𝑚 (see [46] o he explici o m o he ele an oo - inding p oblem), ha is sol ed wi h high accu acy
by means o New on’s me hod. We employ s opping c i e ia in he nonlinea esidual and successi e i e a es o 10−14. I should be
no ed ha he in ini e sum o Eq. (7) is unca ed acco ding o |𝑚|≤𝑘 whe e 𝑘= 41, and hus 2𝑘+ 1 = 83 Fou ie modes we e
employed.
Wa e Mo ion 137 (2025) 103547
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M.L. Ly le e al.
Fig. 1. (Colo online) The spa io- empo al e olu ion o KM b ea he s in he AL and DNLS models [c . Eq. (3) o 𝑔= 1 and 𝑔= 0, espec i ely] wi h 𝑞=𝜔𝑏= 0.005
in Eq. (4). Panel (a) compa es he e olu ion o he ampli ude a he 𝑛= 0 si e, i.e., |𝜓0(𝑡)| wi h solid ed (AL) and blue (DNLS) lines (see, he legend he ein). Panels
(b) and (c) showcase he spa io- empo al e olu ion o he ampli ude |𝜓𝑛(𝑡)| o he KM solu ion in he AL and DNLS models, espec i ely. A la ice o 𝑁= 600
si es was used wi h pe iodic bounda y condi ions.
Fig. 2. (Colo online) Summa y o esul s o he nume ically exac , KM solu ion o he DNLS wi h 𝑞=𝜔𝑏= 0.005. The con e ged KM p o ile is p esen ed in
panel (a) whe e he spa ial dis ibu ion o i s ampli ude is shown. The espec i e Floque s abili y analysis esul s a e p esen ed in panel (b) whe e a eal ye
e y weakly uns able mode exis s wi h magni ude ≈ 1.0000086 (da a no shown). The spa io- empo al e olu ion o he p o ile o panel (a) is shown in panel (c)
showcasing i s obus ness o e 20 pe iods. Finally, he maximum di e ence o he ampli udes a each ime ins an be ween he exac KM solu ion o Eq. (4) and
nume ically e ol ed KM o he DNLS equa ion is shown in panel (d).
Upon using he exac KM b ea he o Eq. (4) as an ini ial guess o New on’s me hod, ou nonlinea sol e con e ged o he p o ile
shown in Fig. 2(a). We pe o med a Floque s abili y analysis o his solu ion ollowing he se up o he a ia ional equa ions o he
associa ed monod omy ma ix as discussed in [46]. The espec i e esul s a e shown in Fig. 2(b) whe e he ed do s co espond o
he Floque mul iplie s 𝜆=𝜆𝑟+i𝜆𝑖. We ind ha all he mul iplie s lie on he uni ci cle (depic ed wi h a solid black line) excep o
a eal eigen alue wi h a iny posi i e eal pa (da a no shown) associa ed wi h a eal ins abili y. Pe he gi en la ice o 𝑁= 600
Wa e Mo ion 137 (2025) 103547
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M.L. Ly le e al.
Fig. 3. (Colo online) MI analysis o he CW solu ion o he DNLS o 𝑞=𝜔= 0.005. In pa icula , eal uns able Floque mul iplie s, i.e., 𝜆𝑟>1 (and 𝜆𝑖= 0)
a e p obed as a unc ion o he numbe o la ice si es 𝑁. The i s uns able mode o CW solu ions appea s when 𝑁 > 628, and mo e and mo e uns able modes
eme ge wi h inc easing 𝑁.
si es and empo al disc e iza ion wi h 83 Fou ie modes, his ins abili y appea s o be a he weak as i s magni ude is ≈ 1.0000086.
We check his inding in Fig. 2(c) which showcases he spa io- empo al e olu ion o he ampli ude |𝜓𝑛(𝑡)| o he nume ically exac
KM b ea he o Fig. 2(a) o e 20 pe iods. The solu ion i sel is qui e obus , main aining i s gene al cha ac e is ics o e he ime
in e al o in eg a ion o he DNLS we conside ed. We pe o m a compa ison be ween he e olu ion o he exac KM b ea he o
Eq. (4) and he nume ically ob ained one o he DNLS model in Fig. 2(d). In pa icula , we moni o he maximum di e ence (in
i s absolu e alue) o he ampli ude o he exac KM solu ion and nume ically exac KM one o he DNLS a each ime ins an . I
can be disce ned om he panel ha despi e being ime-pe iodic solu ions o di e en models hemsel es, i.e., AL and DNLS, hey
a e qui e p oximal o one ano he (no ice he o de o he in ini y no m o he di e ence, i.e., ∼ 10−5).
Finally, we would like o commen on an impo an obse a ion o ou spec al s abili y analysis esul s. I is known om [55],
ha plane wa e solu ions (CW) o he ocusing DNLS model a e modula ionally uns able (a s abili y calcula ion o CW solu ions o
he Sale no model ha con ains he AL is p esen ed in [46] oo). The KM b ea he s (alongside wi h P and AB ones) a e moun ed
a op a ini e ye modula ionally uns able backg ound (pe he ocusing DNLS and AL) whe e localized solu ions may pe u b he CW
Floque spec um. In ligh o he spec al s abili y analysis esul s o Fig. 2(b) and he disc e iza ion used in he p esen calcula ion,
ema kably, no uns able modes a e obse ed o he han he weakly uns able one men ioned p e iously. We also no e ha he
ime- ansla ion mode 1 + 0i is accu a ely esol ed in ou compu a ions.
We in es iga ed he absence o ins abili ies o he KM b ea he o Fig. 2(a) by pe o ming a modula ional ins abili y (MI) analysis
o he CW wi h 𝑞=𝜔𝑏= 0.005 pa ame ically as a unc ion o he numbe o si es 𝑁. Ou espec i e esul s a e shown in Fig. 3 whe e
we p obe he eal uns able modes 𝜆𝑟 as a unc ion o 𝑁. In conjuc ion wi h he spec um o Fig. 2(b) wi h 𝑁= 600, he MI analysis
o he same 𝑁, su p isingly does no p edic he p esence o CW uns able modes. In ac , his is ue o all 𝑁 ≲ 628. Howe e , o a
alue o 𝑁 jus abo e 𝑁= 628, we obse e he eme gence o he i s uns able mode, and in gene al, he numbe o uns able modes
s a s ge ing inc eased wi h 𝑁 [c . Fig. 3], i.e., we ob ain a band o uns able modes as we app oach he in ini e la ice. This inding
in u n sugges s ha o solu ions si ing a op a ini e ye small backg ound, he compu a ion o hei Floque spec um equi es
he use o a e y la ge numbe o si es o esol ing i well. Despi e he absence o a band o eal uns able modes in he Floque
spec um o he KM b ea he o Fig. 2(b) and in he spec a p esen ed nex , his obse a ion indica es ha he KM solu ions ha
we iden i y in his wo k a e expec ed o be MI uns able as 𝑁 inc eases.
Ha ing p esen ed ou p elimina y nume ical esul s o mo i a e he no ion o p oximi y be ween he AL and DNLS models o
small enough da a, we analy ically explo e his p oximi y in he nex sec ion.
4. Theo e ical jus i ica ion o he p oximal KM dynamics be ween AL and DNLS
4.1. The case o he in ini e la ice
Fo ou subsequen heo e ical analysis, we will in oduce no a ion o dis inguish he AL and DNLS limi s o he Sale no model
o Eq. (1). Recall ha he AL model co esponds o he 𝑔= 1 case:
i
𝛹𝑛+𝐶(𝛥𝑑𝛹)𝑛+(𝛹𝑛+1 +𝛹𝑛−1)|𝛹𝑛|2= 0,(8)
whe eas he DNLS model o he 𝑔= 0 case:
i
𝛷𝑛+𝐶(𝛥𝑑𝛷)𝑛+ 2|𝛷𝑛|2𝛷𝑛= 0,(9)
ha is, 𝛹 and 𝛷 will s and o he solu ions o he AL and DNLS models, espec i ely. Mo i a ed by [53], he sys ems will be
supplemen ed wi h gene al non-ze o bounda y condi ions o he o m
lim
|𝑛|→∞𝛹𝑛(𝑡) = lim
|𝑛|→∞𝑒2𝑖𝑞2𝑡𝑞𝑛,lim
|𝑛|→∞𝛷𝑛(𝑡) = lim
|𝑛|→∞𝑒2𝑖𝑞2𝑡𝑞𝑛, 𝑡 ≥0,(10)
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whe e
𝑞𝑛={𝑞−, 𝑛 ≤0,
𝑞+, 𝑛 > 0,(11)
and 𝑞± a e complex cons an s wi h |𝑞±|=𝑞. The bounda y condi ions o Eqs. (10)–(11) can be ans o med o be ime independen
ia he change o a iables
𝛹𝑛(𝑡) = 𝑒2𝑖𝑞2𝑡𝜓𝑛(𝑡), 𝛷𝑛(𝑡) = 𝑒2𝑖𝑞2𝑡𝜙𝑛(𝑡).(12)
Using he change o a iables in Eq. (12), we can ew i e Eqs. (8) and (9) in he o m
𝑖 𝜓𝑛+𝐶(𝛥𝑑𝜓)𝑛− 2𝑞2𝜓𝑛+|𝜓𝑛|2(𝜓𝑛+1 +𝜓𝑛−1)= 0,(13)
𝑖
𝜙𝑛+𝐶(𝛥𝑑𝜙)𝑛+ 2 [|𝜙𝑛|2−𝑞2]𝜙𝑛= 0.(14)
No e ha he ini ial condi ions emain unchanged unde he change o a iables o Eq. (12), namely, 𝜓𝑛(0) = 𝛹𝑛(0) and 𝜙𝑛(0) = 𝛷𝑛(0).
The bounda y condi ions become
lim
𝑛→±∞ 𝜓𝑛(𝑡) = lim
𝑛→±∞ 𝜙𝑛(𝑡) = 𝑞±,(15)
so ha lim|𝑛|→∞|𝜓𝑛(𝑡)|= lim|𝑛|→∞|𝛷𝑛(𝑡)|=𝑞 > 0. Nex , we pe o m a second change o a iables,
𝜓𝑛(𝑡) = 𝑈𝑛(𝑡) + 𝑞𝑛, 𝜙𝑛(𝑡) = 𝑉𝑛(𝑡) + 𝑞𝑛,(16)
and Eqs. (13)–(14) become
𝑖
𝑈𝑛+𝐶(𝛥𝑑𝑈)𝑛+𝐶(𝛥𝑑𝑞)𝑛− 2𝑞2(𝑈𝑛+𝑞𝑛)
+|𝑈𝑛+𝑞𝑛|2(𝑈𝑛+1 +𝑈𝑛−1 +𝑞𝑛+1 +𝑞𝑛−1)= 0,(17)
𝑖
𝑉𝑛+𝐶(𝛥𝑑𝑉)𝑛+𝐶(𝛥𝑑𝑞)𝑛+ 2 [|𝑉𝑛+𝑞𝑛|2−𝑞2](𝑉𝑛+𝑞𝑛) = 0.(18)
The ini ial condi ions o Eqs. (17)–(18), become
𝑈𝑛(0) = 𝜓𝑛(0) − 𝑞𝑛, 𝑉𝑛(0) = 𝜙𝑛(0) − 𝑞𝑛,(19)
and he bounda y condi ions a e ze o a in ini y, i.e.,
lim
𝑛→±∞ 𝑈𝑛(𝑡) = lim
𝑛→±∞ 𝑉𝑛(𝑡) = 0, 𝑡 ≥0.(20)
Along he lines o [56] (see also [53] o he con inuous coun e pa ), we can p o e ha , o any ini ial condi ions 𝑈(0), 𝑉 (0) ∈ 𝓁2,
he e exis 𝑇∗
𝐴𝐿(𝑈(0)), 𝑇 ∗
𝐷𝑁𝐿𝑆 (𝑉(0)) >0 such ha he abo e Cauchy p oblems o he modi ied AL and DNLS Eqs. (17) and (18) ha e
unique solu ions 𝑈∈𝐶1([0, 𝑇 ∗
𝐴𝐿(𝑈(0))],𝓁2) and 𝑉∈𝐶1([0, 𝑇 ∗
𝐷𝑁𝐿𝑆 (𝑉(0))],𝓁2). S a ing om his local exis ence esul and ollowing
he me hods o [44,45,53], we can u he p o e ha hese solu ions s ay close o one ano he in he ollowing sense:
Theo em 4.1. Conside he Cauchy p oblems o he modi ied AL and DNLS models gi en by Eqs. (17) and (18) when supplemen ed wi h
he ini ial condi ions (19) and he anishing bounda y condi ions a in ini y (20). Le any 0< 𝜀 < 1 and assume ha he ini ial condi ions
𝑈(0) and 𝑉(0) ha e 𝓁2-no ms o (𝜀), he 𝓁2-dis ance ‖𝑈(0) − 𝑉(0)‖𝓁2 be ween hem is o (𝜀3), and ha he backg ound ampli ude 𝑞 is
o (𝜀), namely he e exis cons an s 𝐶𝑖, 𝑖 = 1,…,4
‖𝑈(0)‖𝓁2≤𝐶1𝜀, ‖𝑉(0)‖𝓁2≤𝐶2𝜀, ‖𝑈(0) − 𝑉(0)‖𝓁2≤𝐶3𝜀3,and 𝑞≤𝐶4𝜀. (21)
Then, he e exis s 𝑇𝑐>0 and a posi i e cons an 𝐶 > 0 such ha he 𝓁2-dis ance be ween he solu ions sa is ies simul aneously he uppe
bounds
‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2≤𝐶𝜀3𝑡,
‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2≤
𝐶𝜀, o all 0< 𝑡 ≤𝑇𝑐.(22)
Theo em 4.1 is p o ed in Appendix. The ime 𝑇𝑐 in he abo e p oximi y esul is ob ained as ollows. Since 𝑈∈𝐶1([0, 𝑇 ∗
𝐴𝐿],𝓁2),
i.e., i is con inuously di e en iable wi h espec o ime, he assump ion (21) implies ha he e exis s a ime
𝑇𝐴𝐿 ∈ (0, 𝑇 ∗
𝐴𝐿] such
ha he size o 𝑈(𝑡) is also o (𝜀), ha is
‖𝑈(𝑡)‖𝓁2≤𝐶5𝜀, o all 𝑡∈ [0,
𝑇𝐴𝐿],(23)
o some cons an 𝐶5>0. Simila ly, he e is ime
𝑇𝐷𝑁𝐿𝑆 ∈ (0, 𝑇 ∗
𝐷𝑁𝐿𝑆 ] such ha
‖𝑉(𝑡)‖𝓁2≤𝐶6𝜀, o all 𝑡∈ [0,
𝑇𝐷𝑁𝐿𝑆 ],(24)
o some cons an 𝐶6>0. Then, he p oximi y ime 𝑇𝑐 is de ined as 𝑇𝑐= min{
𝑇𝐴𝐿,
𝑇𝐷𝑁𝐿𝑆 }.
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4.2. The case o pe iodic bounda y condi ions
As desc ibed in Sec ion 3, in he nume ical simula ions we supplemen he la ice (3) wi h pe iodic bounda y condi ions. This
scena io bea s some signi ican di e ences and is mo e ac able han he case o he in ini e la ice discussed abo e, due o he con-
se a ion laws o he sys ems. The phase spaces o he pe iodic la ice a e he spaces o pe iodic sequences wi h pe iod 𝑁, deno ed by
𝓁𝑝
𝑝𝑒𝑟 ∶= {𝑈= (𝑈𝑛)𝑛∈Z∈R∶𝑈𝑛=𝑈𝑛+𝑁,‖𝑈‖𝓁𝑝
pe ∶= (ℎ
𝑁−1
∑
𝑛=0 |𝑈𝑛|𝑝)1
𝑝<∞},1≤𝑝≤∞,
whe e ℎ deno es he la ice spacing. Fo simplici y, we se ℎ= 1 (as in he nume ical s udy which co esponds o he choice
𝐶=1
ℎ2= 1 made abo e).
We begin wi h he ollowing analog o Theo em 4.1, which can be p o ed in exac ly he same way as he esul s gi en in [44,45].
Theo em 4.2. Conside he Cauchy p oblems o Eqs. (13) and (14) when supplemen ed wi h pe iodic bounda y condi ions. Le any
0<𝜀<1 and assume ha he ini ial condi ions 𝜓(0) and 𝜙(0) ha e 𝓁2
𝑝𝑒𝑟-no ms o (𝜀), he 𝓁2
𝑝𝑒𝑟-dis ance ‖𝜓(0) − 𝜙(0)‖𝓁2
𝑝𝑒𝑟
be ween hem
is o (𝜀3), i.e. he e exis posi i e cons an s 𝐶𝑖, 𝑖= 1,…,4 such ha
‖𝜓(0)‖𝓁2
𝑝𝑒𝑟 ≤𝐶1𝜀, ‖𝜙(0)‖𝓁2
𝑝𝑒𝑟 ≤𝐶2𝜀, ‖𝜓(0) − 𝜙(0)‖𝓁2
𝑝𝑒𝑟 ≤𝐶3𝜀3.(25)
Then, o a bi a y 0< 𝑇 < ∞, he e exis s a cons an
𝐶 > 0 such ha he 𝓁2
𝑝𝑒𝑟-dis ance be ween he solu ions sa is ies he es ima e
‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2
𝑝𝑒𝑟 ≤
𝐶𝜀3𝑡, o all 0< 𝑡 ≤𝑇 . (26)
The main di e ence be ween Theo ems 4.1 and 4.2 is ha in he la e case he conse a ion laws ensu e he global exis ence
o solu ions, hus implying he alidi y o he p oximi y esul (26) o a bi a y imes. In ac , hanks o global exis ence, we also
ha e he ollowing co olla y o Theo em 4.1, which will be used o jus i y heo e ically he esul s o he nume ical simula ions
shown in Fig. 5 o he case o he non- anishing bounda y condi ions a in ini y:
Co olla y 4.1. Conside he modi ied AL and DNLS models (17) and (18) supplemen ed wi h he ini ial condi ions (19) and pe iodic
bounda y condi ions. Le any 0< 𝜀 < 1 and assume ha he ini ial condi ions 𝑈(0) and 𝑉(0) ha e 𝓁2
pe -no ms o (𝜀), he 𝓁2
pe -dis ance
‖𝑈(0) − 𝑉(0)‖𝓁2
pe
be ween hem is o (𝜀3), and ha he backg ound ampli ude 𝑞 is o (𝜀), namely he e exis cons an s 𝐶𝑖, 𝑖 = 1,…,4
‖𝑈(0)‖𝓁2
pe ≤𝐶1𝜀, ‖𝑉(0)‖𝓁2
pe ≤𝐶2𝜀, ‖𝑈(0) − 𝑉(0)‖𝓁2
pe ≤𝐶3𝜀3,and 𝑞≤𝐶4𝜀. (27)
Then, o a bi a y ini e 0< 𝑇 < ∞, he e exis s posi i e cons an 𝐶 > 0 such ha he 𝓁2
pe -dis ance be ween he solu ions sa is ies
simul aneously he uppe bounds
sup
𝑡∈[0,𝑇 ]‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2
pe ≤𝐶𝜀3.(28)
The p oo o Co olla y 4.1 is iden ical o he one o Theo em 4.1 gi en in Appendix, he only di e ence being ha he no m
used is ha o 𝓁2
pe ins ead o 𝓁2.
4.3. Connec ion o he nume ical indings o Sec ion 3
Theo em 4.1 ( o he in ini e la ice wi h nonze o bounda y condi ions) and Theo em 4.2 ( o he ini e la ice wi h pe iodic
bounda y condi ions) jus i y heo e ically ha he dis ance be ween he solu ions o he AL and DNLS equa ions g ows linea ly a
a a e which is a mos o (𝜀3) when he ini ial and backg ound da a a e o (𝜀).
In he case o Theo em 4.1, acco ding o he well-posedness esul s o [53,57], ini ial da a and backg ound o (𝜀) gua an ee
ha he 𝓁2-no m o he solu ions emains o (𝜀) o e a li espan o (𝜀−2) ≃ 𝑇𝑐. Howe e , beyond ha li espan, he solu ions may
no emain o (𝜀) and, he e o e, he e may exis a ime 𝑇∗
𝑐 such ha he dis ance o solu ions may escape om he apezoidal
egion delimi ed by he wo uppe bounds in (22). This scena io is po ayed by he le diag am o Fig. 4.
On he o he hand, in he case o Theo em 4.2 o he pe iodic p oblem, he (𝜀3) g ow h a e o he di e ence o solu ions is
ansien . Indeed, ecalling he conse ed quan i ies (5) and (6), i was shown in [44] ha
‖𝜓(𝑡)‖2
𝓁2
𝑝𝑒𝑟
≤exp(𝑃AL(0)) − 1 ≲ 𝜀2, o all 𝑡≥0,(29)
and, u he mo e, due o he conse a ion o 𝑃DNLS,
𝑃DNLS(𝑡) = ‖𝜙(𝑡)‖2
𝓁2
𝑝𝑒𝑟
=‖𝜙(0)‖2
𝓁2
𝑝𝑒𝑟
=𝑃DNLS(0) ≲ 𝜀2, o all 𝑡≥0.(30)
Hence, by he iangle inequali y,
‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2
𝑝𝑒𝑟 ≤‖𝜓(𝑡)‖𝓁2
𝑝𝑒𝑟 +‖𝜙(𝑡)‖𝓁2
𝑝𝑒𝑟 ≤
𝐶𝜀, o all 𝑡≥0,(31)
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M.L. Ly le e al.
Fig. 4. (Colo online) The beha io o he dis ance acco ding o he heo e ical uppe bounds o Theo ems 4.1–4.2. Le : The case o nonze o bounda y condi ions.
He e, he dis ance o solu ions may inc ease beyond (𝜀) a ini e ime. Righ : The case o pe iodic bounda y condi ions. He e, acco ding o he uni o m bound
o inequali y (31), he dis ance o solu ions is a mos (𝜀) o all imes. I should be no ed ha , in ce ain scena ios, he dis ance cu e (in blue) may be much
close o he 𝑡-axis han wha is depic ed abo e, a leas o an ini ial ime in e al — e.g. see he co esponding g aphs o Fig. 5.
whe e he cons an
𝐶 is independen o 𝑡. Consequen ly, he linea g ow h uppe bound in he p oximi y es ima e (26) is ele an
only o imes such ha
𝐶𝜀3𝑡≤
𝐶𝜀, i.e.
𝑡≤
𝐶
𝐶
1
𝜀2=∶ 𝑇𝑢𝑏.(32)
Tha is, 𝑇𝑢𝑏 de ines he uppe bound o he imes whe e he linea g ow h o he dis ance be ween he solu ions holds. Fo 𝑡 > 𝑇𝑢𝑏,
he dis ance be ween he solu ions should sa is y he uni o m uppe bound (31). This si ua ion is illus a ed in he igh diag am o
Fig. 4.
Unde he (𝜀) smallness condi ions on he ini ial and backg ound da a (21) and (25), Theo em 4.1 o he in ini e la ice
supplemen ed wi h he nonze o bounda y condi ions (10)–(11) and Theo em 4.2 o he pe iodic la ice jus i y ha he DNLS la ice
admi s solu ions o (𝜀) wi h he ollowing p ope ies:
1. They di e ge a mos linea ly in ime om he analy ical solu ions o he AL la ice in e ms o he 𝓁2-me ic, wi h a linea
g ow h a e a mos o (𝜀3), o ini e imes in [0, 𝑇𝑐] in he case o he in ini e la ice (le panel o Fig. 4) o sa is ying a
mos he uppe bound 𝑇𝑢𝑏 gi en by (32) in he pe iodic case ( igh panel o Fig. 4) .
2. Fo all 𝑡≥0, he solu ions emain close o solu ions o he AL la ice– a mos wi hin (𝜀) wi h espec o he 𝓁2-me ic – o
all 𝑡∈ [0, 𝑇𝑐] in he case o he in ini e la ice and o all 𝑡 > 0 in he pe iodic case.
3. Fo he dis ance measu ed in all 𝓁𝑝-me ics wi h 𝑝≥2, due o he embedding
𝓁𝑞⊂𝓁𝑝,‖𝑢‖𝓁𝑝≤‖𝑢‖𝓁𝑞,1≤𝑞≤𝑝≤∞,(33)
i is easonable o expec an e en smalle a e o p oximi y han he one o he 𝓁2-me ic o he imes desc ibed abo e. Fo
example, when 𝑝= ∞, which is ele an o a compa ison o he ampli udes o solu ions, i is heo e ically jus i ied o expec
an imp o ed a e o p oximi y.
In summa y, compa ing he esul s o Theo ems 4.1 and 4.2, in he case o he in ini e la ice he e is a possibili y ha , a some
ini e ime 𝑇∗
𝑐≥𝑇𝑐≃(𝜀−2), he dis ance o solu ions will escape om he apezoidal egion capped by he (𝜀) dashed ho izon al
line (see Fig. 4). This is due o he ac ha , in he case o nonze o bounda y condi ions, i is no known whe he he solu ions a e
uni o mly bounded. On he o he hand, in he case o pe iodic bounda y condi ions, he solu ions indi idually emain bounded by
he ini ial da a o size (𝜀) due o he conse a ion o 𝑃DNLS and 𝑃AL; he e o e, by he iangle inequali y, he dis ance o solu ions
is gua an eed o emain wi hin he apezoidal egion below he (𝜀) dashed ho izon al line a all imes (see Fig. 4). Howe e ,
since we a e in e es ed in applying hese analy ical esul s o he KM solu ions (4), i is c ucial o unde line he ollowing ema ks
ega ding he beha io o he dis ance o solu ions in 𝓁2
pe :
•I is e iden by he change o a iables (16) ha
‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2
pe =‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2
pe .(34)
Due o Co olla y 4.1, he dis ance ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2
pe
app oxima es well he no m ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2 o he solu ions 𝑈(𝑡) and 𝑉(𝑡)
o he modi ied equa ions Eqs. (17) and (18) when supplemen ed wi h he ini ial condi ions (19) and he anishing bounda y
condi ions a in ini y (20). So we expec in he nume ical simula ions, ha ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2
pe
will sa is y he uppe bounds (22)
o Theo em 4.1 o all 0< 𝑡 < 𝑇𝑐, unde he smallness condi ions on he ini ial da a (27).
•Due o Eq. (34), he dis ance ‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2
pe
will also sa is y he uppe bounds (22) o Theo em 4.1 o all 0<𝑡<𝑇𝑐. In
addi ion, due o he middle es ima e o (29), he conse a ion law (30) and he middle es ima e o (31), will sa is y o all
Wa e Mo ion 137 (2025) 103547
9
M.L. Ly le e al.
Fig. 5. Summa y o nume ical esul s o he AL and DNLS models [c . Eqs. (13) and (14)] in conjuc ion wi h ou heo e ical analysis on hei p oximi y. We
e ol e KM ini ial da a wi h (𝑞, 𝜔𝑏, 𝜀) ≃ (0.09,0.01,0.054) and (𝑞, 𝜔𝑏, 𝜀) ≃ (0.34,0.46,0.425) in he op and bo om ows, espec i ely, o e wo pe iods. Panels (a) and
(d), and (b) and (e) depic he spa io- empo al e olu ion o he ampli udes |𝜓𝑛| and |𝜙𝑛|, espec i ely. The panels (c) and ( ) summa ize he compa isons made
be ween he heo e ical es ima es on he p oximi y o he models wi h he nume ically ob ained dis ance (see, he legend and inse s he ein) o he solu ions
a e sub ac ing he backg ound 𝑞 (i.e. co esponding o he modi ied Eqs. (17) and (18)).
𝑡 > 0, he uni o m in ime bound
‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2
𝑝𝑒𝑟 ≤√exp(𝑃AL(0)) − 1 + √𝑃DNLS(0) ∶= ℎpe .(35)
Fo solu ions sa is ying non-ze o bounda y condi ions as he KM (4), ℎpe can be qui e la ge (gi en he exponen ia ion in ol ed in
he ele an exp ession). In o de o ha e ℎpe =(𝜀) as in he las inequali y o (31), we mus assume he smallness condi ions o
(𝜀) o 𝑃AL(0) and 𝑃DNLS(0) o achie e he inal smallness es ima es o (𝜀2) gi en in (29) and (30). Thus, he dashed ho izon al
line in he igh panel o Fig. 4 which is 𝑦=ℎpe becomes quan i a i ely use ul o solu ions on a non- anishing backg ound 𝑞 as
he KM, only when 𝑞 is su icien ly small, and is o (𝜀) only o 𝑞=(𝜀) wi h 𝜀 ≪ 1. The e o e, o he KM solu ions, he scena io
depic ed in he le panel o Fig. 4 is he one which is quan i a i ely use ul o be explo ed o small and mode a e alues o 𝑞 < 1.
I is ema kable ha he nume ical indings o Sec ion 3 o he case o he KM solu ions showcase ha he g ow h a e o he
dis ance is signi ican ly lowe han he ones associa ed wi h he heo e ical bounds, and he o de o p oximi y is much highe han
wha is p edic ed heo e ically. This is in acco dance wi h se e al nume ical indings o bo h disc e e [44,45] and con inuous [53]
se ups, which illus a e ha he a e o di e gence and p oximi y may depend on he pa icula analy ical solu ion o he in eg able
sys em conside ed. In many cases (b igh soli ons [44], as soli on collisions [53]), his a e is o (𝜀𝑝) wi h 𝑝 > 3. Indica i ely,
and in line wi h he nume ical s udy o Fig. 1, he a e o di e gence o he solu ions was ound o be in he neighbo hood o 𝜀5.
We conclude ou heo e ical analysis by showcasing wo examples compa ing he heo e ical es ima es wi h he nume ical
compu a ions in Fig. 5. No e ha , o compu a ional pu poses, we may conside he assump ions (21) wi h cons an s 𝐶𝑖= 1,
𝑖= 1,…,4. Then, we may conside he ele an imes
𝑇𝐴𝐿,
𝑇𝐷𝑁𝐿𝑆 such ha he bounds (23) and (24) hold wi h he cons an s
𝐶5=𝐶6= 1. Fo his choice o uni cons an s 𝐶𝑖, we may de i e an explici o m o he closeness es ima e (22) by inse ing he
es ima es (23) and (24) in (A.11) and (A.12), espec i ely. This way, by using (A.15), we de i e he ollowing explici o m o he
bound (22) depending on 𝜀 and 𝑞,
‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2≤𝛼𝑡, 𝛼 = 3𝜀3+ 9𝑞𝜀2+ 12𝑞2𝜀. (36)
The op and bo om ows o Fig. 5 co espond o he cases wi h (𝑞, 𝜔𝑏) = (0.09,0.01) and (0.34,0.46), espec i ely. Same as be o e,
we employ a la ice o 𝑁= 600 si es, and use he exac KM solu ion o Eq. (4) as an ini ial condi ion o bo h he AL and DNLS
models. The panels (a) and (d), and (b) and (e) showcase he spa io- empo al e olu ion o he ampli ude o he KM solu ion (in
he espec i e cases) o he AL and DNLS models, espec i ely, o e 2 pe iods. In line wi h Fig. 4, we summa ize ou p oximi y
esul s in panels (c) and ( ) o Fig. 5 whe e he solid blue line (see he legend in panel (c)) shows he dis ance o he solu ions in
he 𝓁2-no m a e emo ing he backg ound 𝑞. The ho izon al dashed black line co esponds o 𝑦= 2‖𝑈(0)‖𝓁2= 2𝜀, whe eas he
solid ed line o 𝑦=𝛼𝑡, wi h 𝛼 gi en by (36). Fo panels (a)–(c), 𝜀≃ 0.054, while o panels (d)–( ) we used 𝜀≃ 0.425. Recall ha ,
acco ding o he heo e ical esul s o Theo em 4.1 and Co olla y 4.1, he solu ions should emain p oximal o minimal gua an eed
imes o 𝑇𝑐≃(𝜀−2). The nume ical esul s o Fig. 5(c) and ( ) con i m his ac , since hey show ha he wo models a e p oximal
up o imes ≈ 512 ( op ow) and ≈ 5.86 (bo om ow), see he inse s he ein. Fu he mo e, he compa ison o he pa e ns o e he
Wa e Mo ion 137 (2025) 103547
16
M.L. Ly le e al.
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