Summary_Akhmedova (1137473)
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National Research University Higher School of EconomicsFaculty of Mathematicsas a manuscriptValeriya AkhmedovaReductions of dispersionless integrable hierarchies andLowner equationSummary of the PhD thesisfor the purpose of obtainingPhilosophy Doctor in Mathematics HSEAcademic supervisorAnton Zabrodin Doctor ofScience, professorMoscow 2018IntroductionIt is well-known that parametric families of univalent conformal mappingsof domains with a cut along some curve on xed canonical domains (usuallyan upper half-plane or a unit disk) are described with the help of dierentialLowner equation. We begin our study from the history of this equation.The ordinary dierential Lowner equation is given by one-parameterfamily of conformal mappings of canonical regions into themselves and servesas a powerful tool to study properties of univalent functions.
For the rsttime it appeared in the work of Charles Lowner in 1923 and was writtenfor functions dened in the unit disk D. The equation contains an arbitrarymeasurable function that plays the role of a "driving" function. Later, innew versions of the Lowner equation, other canonical domains were considered:a half-plane, a strip, a ring. The greatest attention in recent years has beengiven to the "radial" equation for D and the "chordal" the equation for theupper half-plane H.As was shown in the works of Gibbons and Tsarev, there is an interestingconnection between Lowner equations and integrable hierarchies of nonlinearpartial dierential equations. The chordal Lowner equation plays a key rolein classifying reductions of the KP hierarchy in dispersionless limit. Namely,it is the consistency condition for the one-component reduction with thewhole innite hierarchy.
The radial Lowner equation plays analogical role forthe hierarchy of the dispersionless two-dimensional Toda chain. The easiestway to see the connection between dispersionless hierarchies and the Lownertype equation is to consider the hierarchy of Kadomtsev-Petviashvili (KP).The dispersionless KP equation is as follows33(ut − uux )x − uyy = 0.24(1)The dispersionless hierarchy of Kadomtsev-Petviashvili (dKP) is a an innitesystem of nonlinear dierential equations.This hierarchy is well studied, and it has several equivalent representations,however, to illustrate the connection with the Lowner equation in this work,we will use only the corresponding Hirota equation:eD(z)D(ζ)F = 1 −∂t1 (D(z) − D(ζ))F.z−ζ(2)We show the connection with the Lowner equation by considering onecomponent reductions of this hierarchy.2After illustrating the emerging connection, we will proceed to investigatingdispersionless hierarchies of Pfa-KP and Pfa-Toda.The DKP hierarchy is one of the integrable hierarchies with D∞ symmetriesintroduced by M.Jimbo and T.Miwa in 1983.
It was subsequently rediscoveredand came to be also known as the coupled KP hierarchy and the Pfa lattice.The latter name is motivated by the fact that some solutions to the hierarchyare expressed through Pfaans.Although in this thesis we study only dispersionless hierarchies, it isuseful, however, to look at the "full" hierarchy to see what happens when wetake dispersionless limit. Thus, the rst equation of the Pfa-KP hierarchy,the so-called DKP equation, looks as follows: ∂∂u ∂t1 −4 ∂t3 +2 ∂v± +∂t3∂ 3 v±∂t31∂3u∂t31++∂u12u ∂t1±6u ∂v∂t1∓322+ 3 ∂∂tu2 = 12 ∂∂tu2 (v + v − )2∂ 2 v±∂t1 ∂t2+ 2v1±R∂u∂t2 dt1= 0.As one can see, the left hand side of the rst equation is just KP equation,and the right hand side gives a coupling term with the eld v ± . Because ofthis, the DKP equation is sometimes called a coupled KP equation (cKP).In terms of the τ -functions, u and v ± are expressed as∂2τn±1u = 2 log τn , v ± =.∂t1τnWe can use Hirota operators (Hirota derivatives), which are dened asfollowsDk f · g :=∂∂− 0∂tk ∂tkf (tk )g(t0k ).tk =t0kWith their help, the DKP equation is given((−4D1 D3 + D14 + 3D22 )τn τn = 24τn+1 τn−1(2D3 + D13 ∓ 3D1 D2 )τn±1 τn = 0.(3)Bearing certain similarities with the KP and Toda chain hierarchies, theDKP one is essentially dierent and less well understood.The dispersionless version of the DKP hierarchy (the dDKP hierarchy)was suggested by Takasaki.
It is an innite system of dierential equationsfor a real-valued function F = F (t) of the innite number of (real) times3t = {t0 , t1 , t2 , . . .}. The dierential equations are obtained by expandingequations1 2∂t (2∂t +D(z)+D(ζ))F∂t D(z)F − ∂t1 D(ζ)FD(z)D(ζ)Fe,1− 2 2e 0 0=1− 1z ζz−ζ(4)2 −2∂t0 D(z)F2 −2∂t0 D(ζ)Fz e−ζ ee−D(z)D(ζ)F= z +ζ −∂t1 2∂t0 +D(z)+D(ζ) F,z−ζ(5)whereX z −k∂tk(6)D(z) =kk≥1in powers of z , ζ . The function F corresponds to the logarithm of thetau function in the case of the dispersionless KP hierarchy. The dierentialequations are obtained by expanding equations (4), (5) in powers of z , ζ .For example, the rst two equations of the hierarchy are2+ 3F22 − 4F13 = 12e4F00 6F112F03 +34F01(7)+ 6F01 F11 − 6F01 F02 = 3F12 .We use the short-hand notation Fmn ≡ ∂tm ∂tn F .We could obtain the same result directly from the DKP equation (3) byusing v ± = exp(log τn±1 − log τn ) → exp(±h̄−1 F0 ) è v + v − = exp(log τn+1 −2 log τn + log τn−1 ) → exp(F00 ) ïðè h̄ → 0.The 2D Pfa-Toda hierarchy suggested by Willox and Takasaki is anextension of the Pfa-KP hierarchy which relates to it in the same way as the2DTL relates to the KP hierarchy.
In particular, the extension Pfa-KP −→Pfa-Toda implies doubling of the set of hierarchical times. Here we deal withreal forms of the hierarchies which means that the KP times are assumedto be real while the two sets of Toda times are complex conjugate to eachother.The dispersionless version of the Pfa-Toda hierarchy (dPfa-Toda) iswritten for a function F of the doubly-innite set of times {. .
. , t̄2 , t̄1 , r, s,t1 , t2 , . . .}. Since the dierent hierarchies are never mixed in this paper, wekeep the same notation F for the dispersionless tau-function. The real formof the hierarchy, which we will be dealt with, implies that t̄k is complexconjugate to tk , s is real and r is purely imaginary. The basic equations are4as follows:1 ∂seD(z)D(ζ)F 1 −ezζ ∂s +∂r +D(z)+D(ζ) F1 ∂seD̄(z̄)D̄(ζ̄)F 1 −ez̄ ζ̄1 ∂reD(z)D(ζ)F 1 −ezζ= ∂s −∂r +D̄(z̄)+D̄(ζ̄) Fze−∂r D(z)F − ζe−∂r D(ζ)F, (8)z−ζz̄e∂r D̄(z̄)F − ζ̄e∂r D̄(ζ̄)F=,z̄ − ζ̄(9)ze−∂s D(z)F − ζe−∂s D(ζ)F=,z−ζ(10) −∂s D̄(z̄)F−∂s D̄(ζ̄)Fz̄e− ζ̄e1 −∂r ∂s −∂r +D̄(z̄)+D̄(ζ̄) F=e,eD̄(z̄)D̄(ζ̄)F 1 −z̄ ζ̄z̄ − ζ̄(11) 1 ∂s ∂s +D(z)+D̄(ζ̄) F1 ∂r ∂r +D(z)−D̄(ζ̄) F=1−ee, (12)e−D(z)D̄(ζ̄)F 1 −z ζ̄z ζ̄ z −∂r ∂s +D(z)+D̄(ζ̄) F − ∂s −∂r +D̄(ζ̄) D(z)F− ∂s +∂r +D(z) D̄(ζ̄)Fe−1= ee−1 .ζ̄(13)X z̄ −k∂t̄k is the complex conjugate counterpart of the dierentialHere D̄(z̄) =k ∂s +∂r +D(z)+D(ζ) Fk≥1operator (6). Note that equations (9), (11) are obtained from, respectively,(8), (10) by applying the bar-operation D → D̄, z → z̄ , ζ → ζ̄ , tk → t̄k ,s → s̄ = s, r → r̄ = −r which can be treated as complex conjugationprovided the function F is real.
We see that each equation has a barcounterpart. At the same time, the other two equations, (12) and (13),are real, i.e. they do not change under the complex conjugation. From nowon we will not always write explicitly the conjugates of complex equationskeeping in mind that they hold simultaneously. In what follows it will bemore convenient to introduce the complex conjugate 0th times t0 = s + r,t̄0 = s − r, so ∂t0 = 21 (∂s + ∂r ), ∂t̄0 = 12 (∂s − ∂r ).The dierential equations are obtained by expanding (8)(13) in powersof z , ζ , z̄ , ζ̄ . The two simplest equations of the hierarchy are F e 00 F01̄ = eF0̄0̄ F0̄1 ,F00 +F0̄0̄F11̄ = 2 e(14)sinh 2F00̄ .Here Fmn ≡ ∂tm ∂tn F , Fmn̄ ≡ ∂tm ∂t̄n F , Fm̄n̄ ≡ ∂t̄m ∂t̄n F .5Chapter 2In this chapter we give a short historical review about Lowner equationsand demonstrate exactly how one of them arises in the context of integrablehierarchies.
For this porpose we consider a dispersionless hierarchy of KadomtsevPetviashvili. The Hirota equation for this hierarchy has the form:eD(z1 )D(z2 )F =z1 − z2 − ∂t1 D(z1 )F + ∂t1 D(z2 )F.z1 − z2(15)We introduce the functionp(z) = z − ∂t1 D(z)F.(16)The function z(p) is inverse to p(z), we consider it to be univalent nearinnity.u1 u2+ 2 + ...ppThe coecients ui depend on the real parameters t1 , t2 , t3 . . .
("times"). Weconsider the following reduction - let z(p) depend on all times tj throughonly one function U = U ({tj }), i.ez(p) = p +z(p; {tj }) = z(p, U ),U = U ({tj })The condition for the consistency of a one-component reduction with ahierarchy is equation∂p(z)1=−.∂Up(z) − ξ(U )(17)An analogous equation for z(p) has the form:∂z(p)1∂z(p)=.∂Up − ξ(U ) ∂p(18)Thus, we have obtained the chordal Lowner equation with a driving functionξ(U ).Chapter 3We consider in detail the Pfaan hierarchies in the algebraic formulation,then we proceed to an elliptical formulation of a dispersion-free hierarchy of6Pfa-KP, and generalize the result to the case of a hierarchy of dispersionlessPfa-Toda. In this work we show that the Pfa-type hierarchies admit a nicereformulation in terms of elliptic functions (or Jacobi theta functions).
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