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Afterthis reformulation, the number of independent equations gets reduced andsomewhat unsightly looking equations (4), (5) and especially (8)(13) assumecompact and suggestive forms which look like natural elliptic deformationsof the dispersionless KP (or modied KP) and 2DTL hierarchies.We show that equations (4), (5), when rewritten in an elliptic parametrization in terms of Jacobi's theta-functions θa (u, τ ), assume a niceand suggestive form which looks like a natural elliptic extension of thedispersionless KP hierarchy:(z −1 − ζ −1 )e(∂t0 +D(z))(∂t0 +D(ζ))F =θ1 (u(z)−u(ζ), τ ).θ4 (u(z)−u(ζ), τ )(19)Here the function u(z) is dened bye∂t0 (∂t0 +D(z))F = zθ1 (u(z), τ ).θ4 (u(z), τ )(20)The modular parameter τ is a dynamical variable: τ = τ (t).

We assumethat τ is purely imaginary. In what follows we use the dierential operator∇(z) = ∂t0 + D(z)(21)which appears to be more convenient than D(z).To give yet another instructive form of equation (19), it is convenient tointroduce the functionS(u| τ ) := logθ1 (u|τ ).θ4 (u|τ )(22)In terms of the function S(u), the equation (19) reads∇(z1 )S(u(z2 )|τ ) = ∂t0 S(u(z1 ) − u(z2 )|τ ).(23)Certain manipulations and elliptic parametrization allow one to represent7the equations of the dispersionless Pfa-Toda hierarchy (8) - (13) in the form(z1−1 − z2−1 ) e∇(z1 )∇(z2 )F =¯e∇(z1 )∇(z̄2 )F =¯¯(z1−1 − z2−1 ) e∇(z1 )∇(z2 )F =θ1 (u(z1 ) − u(z2 ))θ4 (u(z1 ) − u(z2 ))θ1 (u(z1 ) + ū(z2 ) + η)θ4 (u(z1 ) + ū(z2 ) + η)(24)θ1 (ū(z1 ) − ū(z2 ))θ4 (ū(z1 ) − ū(z2 ))Here we used the notation¯∇(z)= ∂t̄0 + D̄(z).(25)The rst equation is the same as (19). This means that a half of thedispersionless Pfa-Toda hierarchy (with xed bar-times) coincides with thePfa-KP one.

This fact can not be so transparently seen in the algebraicformulation. The third equation is the bar-version of the rst one. It representsanother copy of the dPfa-KP hierarchy, now with respect to the bar-times t̄kwith xed tk 's. The second equation contains mixed derivatives with respectto the times {tk } and {t̄k } and thus it couples the two hierarchies into themore general one.

This equation is invariant under complex conjugation.The modular parameter τ is a dynamical variable: τ = τ (t). This featuresuggests some similarities with the genus 1 Whitham equations and theintegrable structures behind boundary value problems in plane doubly-connecteddomains.Since the Pfaan hierarchies are much less studied than the more familiardispersionless KP (dKP), mKP (dmKP) and 2DTL (d2DTL) ones, we foundit useful to give a detailed comparison of these hierarchies.Chapter 4This chapter is devoted to one-component reductions of Pfaan hierarchies.As in the previous chapter, we begin the study of Pfaan hierarchies with asimpler Pfa-KP hierarchy. We investigate one-variable reductions of thishierarchy assuming that all dynamical variables depend on the times tthrough a single variable which in a generic case can be identied withthe modular parameter τ .

We show that such reductions are classied bysolutions of a dierential equation which is an elliptic analogue of the famous8Lowner equation. In complex analysis, this elliptic Lowner equation is alsoknown as the Goluzin-Komatu equation: τ τ4πi ∂τ u(z, τ ) = − ζ1 u(z, τ )+ξ(τ ) 2 + ζ1 ξ(τ ) 2 ,(26)where ζ1 (u, τ ) := ∂u log θ1 (u|τ ) and ξ(τ ) is an arbitrary (continuous) functionof τ (the driving function).

This equation is the basic element of the theoryof parametric conformal maps from doubly connected slit domains to annuli.In order to complete the description of one-variable reductions, we shouldderive the equation satised by τ (t) and nd its solution.S 0 u(z) + ξ(τ )∂t0 τ.(27)S 0 (ξ(τ ))This is a generating equation for a hierarchy of equations of the hydrodynamictype. To write them explicitly, we use the expansion∇(z)τ =00S (u(z) + u) = S (u) +X z −kk≥1kBk0 (u),(28)which denes the functions Bk0 (u) = Bk0 (u|τ ). In terms of these functions,the equations of the reduced hierarchy are as follows:∂τ∂τ= φk (ξ(τ )|τ ),∂tk∂t0φk (ξ(τ )|τ ) :=Bk0 (ξ(τ )|τ ),S 0 (ξ(τ )|τ )k ≥ 1.(29)The common solution to these equations can be written in the hodographform:∞Xtk φk (ξ(τ )|τ ) = Φ(τ ),(30)k=1where Φ(τ ) is an arbitrary function of τ .In the simplest case, when Φ(τ ) =0, we conclude from (30) thatXk≥1tk∂τ= 0, i.e., τ (t) is a homogeneous∂tkfunction of the times of degree 0.Finally, we point out an unexpected connection with the Painleve VIequation.

Namely, we show that the second τ -derivative of the elliptic Lownerequation (26), with a particular choice of the driving function, gives thePainleve VI equation with special values of the parameters written in theelliptic (Calogero-like) form.9Further we make similar calculations for the one-component reductionof dispersionless Pfa-Toda hierarchy.

Our goal is to characterize the classof functions u(z, λ), η(λ), τ (λ) that are consistent with the hierarchy. Forsimplicity, in what follows we put λ = τ .We get that the sucient conditions for the functions u(z, τ ), ū(z, τ ) andη(τ ) to be compatible with the innite dPfa-Toda hierarchy are equations:4πi ∂τ η(τ ) = −ζ1η2 τη+ iκ 2 − ζ1 2 − iκ τ2 τη4πi∂τ u(z, τ ) = −ζ1 u + + iκ 2 + ζ1 2 + iκ τ2η2(31)  4πi∂τ ū(z, τ ) = −ζ1 u + η − iκ τ + ζ1 η − iκ τ2222The generating equations for the innite reduced hierarchy are as follows:¯S 0 (ū(z̄) + ξ)¯∇(z̄)τ = −∂t0 τ.S 0 (ξ)S 0 (u(z) + ξ)∇(z)τ =∂t0 τ ,S 0 (ξ)(32)They are equations of hydrodynamic type.

To write them explicitly, as inthe previous case, we use elliptic analogues of Faber polynomials. We obtainthe equations:∂τ∂τ= φk (ξ(τ )|τ ),∂tk∂t0∂τ∂τ= ψk (ξ(τ )|τ ),∂ t̄k∂t0(33)whereφk (ξ(τ )|τ ) =Bk0 (ξ(τ )|τ ),S 0 (ξ(τ )|τ )ψk (ξ(τ )|τ ) = −¯ )|τ )B̄k0 (ξ(τ.S 0 (ξ(τ )|τ )(34)The common solution to these equations can be represented in the hodographform:XXtk φk (ξ(τ )) +t̄k ψ0 (ξ(τ )) = Φ(τ ).(35)k≥1k≥0Here Φ is an arbitrary function of τ .Chapter 5In this chapter we study diagonal N -variable reductions of the dDKP hierarchywhen u depends on the times through N real variables λj .

The starting10point is the system of N elliptic Lowner equations which characterize thedependence of u(z) on the variables λj :i ∂τττ4πi ∂λj u(z, {λi }) = − ζ1 u+ξj , 2 + ζ1 ξj , 2,∂λjh(36)Their compatibility condition is expressed as the elliptic Gibbons-Tsarevsystem ∂τ1 ∂ξk00=ζ1 (−ξk + ξj , τ ) − ζ1 (ξj , τ ),∂λj4πi∂λj∂ 2τ1∂τ ∂τ=℘1 (ξk − ξj , τ 0 )∂λk ∂λj2πi∂λk ∂λjfor all j = 1, . . . , N , j 6= k .(37)(38)The time dependence of the variables λj is xed by a system of quasilinear partial dierential equations of the form∂λj∂λj= φj,k ({λi }),∂tk∂t0(39)with φj,k ({λi }) dened with the help of elliptic Faber functions. As is easyto see, the compatibility condition of the system (39) is∂λj φi,n∂λj φi,n0=φj,n − φi,nφj,n0 − φi,n0for all i 6= j , n, n0 .In other words, we should show thatΓij :=∂λj φi,nφj,n − φi,n(40)does not depend on n, i.e., (40) holds for all n simultaneously.

Then we showthat the system (39) can be solved by the generalized hodograph methoddeveloped by Tsarev and associated with the system diagonal metric is ofEgorov type.ConclusionIn this thesis we study the dispersionless hierarchies of Pfa-KP and PfaToda, their reductions, and conditions under which reductions are permissible.The main technical tools are the methods of elliptic functions and the theoryof integrable systems.11References1. V. Akhmedova and A.

Zabrodin, Dispersionless DKP hierarchy andelliptic Lowner equation, J. Phys. A: Math. Theor. 47 (2014) 392001.arXiv:1404.51352. V. Akhmedova and A. Zabrodin, Elliptic parametrization of Pfaintegrable hierarchies in the zero dispersion limit, Theor. Math. Phys.185 (2015) 410-422.arXiv:1412.84353. V. Akhmedova and A. Zabrodin, Dispersionless Pfa-Toda hierarchyand elliptic Lowner equation, J. of Math.

Phys. 57-10 (2016).arXiv:1605.015614. V. Akhmedova, T. Takebe, A. Zabrodin, Multi-variable reductions ofthe dispersionless DKP hierarchy, J. Phys. A: Math. Theor. 50 (2017). arXiv:1707.0152812.

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