By Sergey Nazarenko
The widening spectrum of actual functions of wave turbulence thought levels from plasma waves to Bose-Einstein condensates. coming up from lectures given by way of the writer, this quantity is based as a graduate textual content and comprises routines and ideas.
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Additional info for Wave Turbulence (Lecture Notes in Physics)
This step is essential. may still we 90 6 Wave Turbulence Formalism now not have performed this, the next derivation technique wouldn't be selfconsistent. workout 6. 12 outline the interplay illustration variables bk as earlier than, with out xNL and take a look at to persist with the WT derivation steps. At which aspect do you get inconsistency? What actual interpretation are you able to recommend? when it comes to bk we've ib_ okay ¼ 2 X 12 i W 12 3k d3k e R x3k dt 12 b1 b2 bÃ3 ; ð6:76Þ k1 ;k2 6¼k3 the place x3k 12 ¼ xk þ xNL okay þ x3 þ xNL three À x1 À xNL 1 À x2 À xNL 2 . (Note that for NLS the nonlinear frequency might drop out from this expression since it is k-independent). As earlier than, your next step is to contemplate resolution bk(T) at an intermediate (between linear and nonlinear) time 2p=xk ( T ( 2p=4 xk (we should not have to ð0Þ ð1Þ call for xNL T ) 1! ). We search this answer as a sequence bk ðTÞ ¼ bk þ 2 bk þ ð2Þ ð0Þ 4 bk þ Á Á Á got through recursive substitution into (6. 76). this offers bk ¼ bk jT ¼ zero and X ð1Þ 12 ð0Þ ð0Þ ð0ÞÃ 3k W 12 ð6:77Þ bk ðTÞ ¼ Ài 3k d3k b1 b2 b3 DT ðx12 Þ: k1 ;k2 6¼k3 the place functionality DT is outlined in (6. 40). For the second one new release we get X ð2Þ 34 k5 34 k5 ð0Þ ð0Þ ð0Þ ð0ÞÃ ð0ÞÃ k5 W15 W62 d15 d62 b1 b6 b2 b3 b4 Eðx34 bk ðTÞ ¼ 15 ; x62 Þ 16¼3;4;56¼6;2 sixteen k5 sixteen k5 ð0ÞÃ ð0ÞÃ ð0Þ ð0Þ ð0Þ k5 À2W34 W62 d34 d62 b1 b5 b3 b4 b2 Eðx16 34 ; x62 Þ ; the place Eðx; yÞ ¼ RT zero ð6:78Þ Dt ðxÞeiyt dt (like in (6. 24)). ð1Þ your next step, back like sooner than, is to discover the section averages of jbk j2 and of ð2Þ ð0ÞÃ bk bk , yet now utilizing the hot expressions (6. seventy seven) and (6. 78). this can be to be by means of substitution into the expression (6. 30), amplitude averaging and taking the large-box and the small- limits. those steps are similar to what we did sooner than and we depart them for the reader as an workout: workout 6. thirteen persist with the stairs defined within the earlier paragraph. exhibit that this ends up in an identical equations for the one-point producing functionality (6. 45), the onepoint PDF ((6. fifty three) and (6. 52)), and the kinetic equation (6. 55), yet now with new expressions for gk and ck: Z four k3 2 k3 gk ¼ 4p j d12 dðxk3 ð6:79Þ jW12 12 Þn1 n2 n3 dk1 dk2 dk3 ; 6. nine Four-Wave and Higher-Order platforms ck ¼ 4p four Z ninety one k3 2 k3 jW12 j d12 dðxk3 12 Þ½n3 ðn1 þ n2 Þ À n1 n2 dk1 dk2 dk3 : ð6:80Þ With those new expressions for gk and ck, the kinetic equation (6. fifty five) turns into ! Z 1 1 1 1 four k3 2 k3 k3 þ À À n_ ok ¼ 4p jW12 j d12 dðx12 Þn1 n2 n3 nk dk1 dk2 dk3 : ð6:81Þ n ok n3 n1 n2 6. nine. 2 structures with Higher-Order Wave Resonances If the reader has rigorously the former sections, she or he has sufficient wisdom and adventure for facing structures characterised via wave resonances with orders larger than 4. essentially, so much paintings has already been performed prior to, and the single factor to be performed in each one specific case is to calculate the 1st ð1Þ and the second one order contributions within the interplay illustration variables, bk ð2Þ ð1Þ ð2Þ ð0ÞÃ and of bk , and to discover the section averages of jbk j2 and of bk bk .