Steady State Solution of the Kuramoto-Sivashinsky PDE J. C. Sprott
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1 Stey Stte Soltio of the Krmoto-Sivshisy PDE J. C. Srott The Krmoto-Sivshisy etio is simle oe-imesiol rtil ifferetil etio PDE tht ehiits hos er some oitios. I its simlest form, the etio is give y t 0 where the ssrits eote ifferetitio of the stte vrile with reset to time se, resetively. Here we see stey stte stig wve soltios t 0 to the etio i ifiite stil omi g Forier lysis. The stility of sh soltios is serte mtter to e emie lter. The simlest moel ists of gle e wve: Derivtives: Nolier term: Simlify g the followig trigoometri ietity: [ To oti the followig: Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term: 0 0
2 Clerly there is o soltio eet for 0 or 0. However, the first etio hs seo soltio give y, whih is ot very ifferet from the vle oserve merilly t 6/
3 slightly more relisti moel, motivte y meril soltio of the KS etio is: Derivtives: 8 6 Nolier term: Simlify g the followig trigoometri ietity: [ To oti the followig: [ [ Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term: This system is over-seifie e there re for etios for three ows. However, the lst two etios re oly roimtios e they re iistet with the ssmtio tht oly terms i re reset. The first two etios re et e simlifie for,, ozero to: 8 From the meril soltio of the KS etio, we hve 6/ , from whih we etermie :
4 These vles re i resole greemet with meril reslts. Hee:
5 ssme iste more geerl moel: There is o loss of geerlity i igorig the term. Derivtives: Nolier term: Simlify g the followig trigoometri ietities: [ [ [ To oti the followig: [ [ [ [ [ [ Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term:
6 Simlify: / 8 ± Elimite /: ± 8 ± Elimite : 6 Solve the seo etio ove for : / ± From, llte,,, : ± ± ± / ± ± 6 ± Hee:
7 7 Strt over t with two etr terms thir hrmoi: f e Derivtives: f e f e f e f e Nolier term: f ef f f f ef e e e e f e f e f e f e Simlify g the followig trigoometri ietities: [ [ [ To oti the followig: f ef f f f ef e e e e f e f e f e f e 6 [6 [ [ [ 6 [ 6 [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ Stey stte of Krmoto-Sivshisi etio: 0
8 Etig term-y-term: e f 0 e 6 0 f 6 0 f 9e 8e 0 e 9 f 8 f 0 e 0 f 0 e f 0 f e The 6 6 terms re igore e they wol reire e f 0. Simlify: [ e f 0 [8 [6e 9 [6 f9 [ f [ e f f e 0 0 f e / 0 / 0 e 0 0 / 0 [ f e/ 0 This system is over-etermie e there re ie etios for seve ows. Ug oly the first seve etios igorig the terms gives the followig et meril reslt my ot e ie: 8
9 e 0.76 f 0 Hee: However, we erform meril lest-sres fit to the etire system of ie etios with the followig reslt me sre error ~ : e f Hee:
10 The revios reslts sggest tht we igore the oe terms: Derivtives: Nolier term: Simlify g the followig trigoometri ietity: [ To oti the followig: [ [ [ [ [ [ 6 Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term: 0
11 Simlify: The oly soltios hve 0 either 0 or ±.
12 Let s oe more term : Derivtives: Nolier term: Simlify g the followig trigoometri ietity: [ To oti the followig: 8 [7 [6 [ [7 6 [ [ [6 [ [ [ [ [ Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term:
13 The 7 8 terms re igore e they wol reire 0. Simlify: [ / 0 [8 [ [ / 0 [ / 0 / 0 / 0 0 We hve si etios five ows. Igorig the lst etio 6 gives the followig et meril soltio ot eessrily ie: Hee: Rell tht the eete vle from the meril soltio of the K-S etio is 6/ The reslts from the Forier esio er to e overgig o tht vle t very slowly The mlite is lso overgig towr lsile vle ~. for the term.
14 Let s oe more term : e Derivtives: e e e e Nolier term: e e e e e e e e Simlify g the followig trigoometri ietity: [ To oti the followig:
15 e e e e e e e e e 0 [9 [8 [7 [6 [9 8 [7 [6 [ [8 [7 6 [ [ [7 [6 [ [ [6 [ [ [ Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term: e e e e e e e e e e e e e e Simlify:
16 [ [8 [ [0 e [e e/ 0 e/ 0 e / 0 e / 0 / 0 0 7[ e / 0 We hve seve etios si ows. Igorig the lst etio 7 gives the followig et meril soltio ot eessrily ie: e Hee: This is the est moel yet, t overgee is very slow. 6
17 7 Let s llte the geerl moel with hrmois, ll i hse: Derivtives: Nolier term: Simlify g the followig trigoometri ietity: [ To oti the followig: Stey stte of Krmoto-Sivshisi etio: 0 Etig term-y-term: 0 Simlify: 0
18 We hve etios to ows,,. Ths for ll > the system is otetilly overetermie. However, for sh olier system, there is o grtee of soltio, whe oe eists, there is o grtee tht it is ie. Ths we ot meril roere of miimizig the vle of ll ftios y lest sres. This mots to emig tht the wveform of resltig from the omitio of the Forier terms whih is el to t is s smll s ossile. tlly, the tity miimize is t / to voi settlig ito the trivil soltio with ll tities zero. s he, t re lotte verss for 0 < < / oe wve of the fmetl wvelegth. Nmeril reslts re s follows:
19 ei These re vles otie y Forier lysis of the stey stte meril soltio of the Krmoto-Sivshisy etio ortesy of Jo Seto: Rel Prt Imgiry Prt mlite Phse
a f(x)dx is divergent.
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