We discuss the recently established equivalence between the Laplacian growth
in the limit of zero surface tension and the universal Whitham hierarchy known
in soliton theory. This equivalence allows one to distinguish a class of exact
solutions to the Laplacian growth problem in the multiply-connected case. These
solutions corerespond to finite-dimensional reductions of the Whitham hierarchy
representable as equations of hydrodynamic type which are solvable by means of
the generalized hodograph method.Comment: 20 pages, 4 figures, based on the talk given at the Workshop
``Classical and quantum integrable systems'' (Dubna, January 2004), minor
correction
Let I(t)=∮δ(t)​ω be an Abelian integral, where
H=y2−xn+1+P(x) is a hyperelliptic polynomial of Morse type, δ(t) a
horizontal family of cycles in the curves {H=t}, and ω a polynomial
1-form in the variables x and y. We provide an upper bound on the
multiplicity of I(t), away from the critical values of H. Namely: $ord\
I(t) \leq n-1+\frac{n(n-1)}{2}if\deg \omega <\deg H=n+1.Thereasoninggoesasfollows:weconsidertheanalyticcurveparameterizedbytheintegralsalong\delta(t)ofthen‘‘Petrov′′formsofH(polynomial1−formsthatfreelygeneratethemoduleofrelativecohomologyofH),andinterpretthemultiplicityofI(t)astheorderofcontactof\gamma(t)andalinearhyperplaneof\textbf C^ n.UsingthePicard−Fuchssystemsatisfiedby\gamma(t),weestablishanalgebraicidentityinvolvingthewronskiandeterminantoftheintegralsoftheoriginalform\omegaalongabasisofthehomologyofthegenericfiberofH.Thelatterwronskianisanalyzedthroughthisidentity,whichyieldstheestimateonthemultiplicityofI(t).Still,insomecases,relatedtothegeometryatinfinityofthecurves\{H=t\}
\subseteq \textbf C^2,thewronskianoccurstobezeroidentically.Inthisalternativeweshowhowtoadapttheargumenttoasystemofsmallerrank,andgetanontrivialwronskian.Foraform\omegaofarbitrarydegree,weareledtoestimatingtheorderofcontactbetween\gamma(t)andasuitablealgebraichypersurfacein\textbf C^{n+1}.Weobservethatord I(t)growslikeanaffinefunctionwithrespectto\deg \omega$.Comment: 18 page
Given a function f holomorphic at infinity, the n-th diagonal Pad\'e
approximant to f, denoted by [n/n]f​, is a rational function of type
(n,n) that has the highest order of contact with f at infinity. Nuttall's
theorem provides an asymptotic formula for the error of approximation
f−[n/n]f​ in the case where f is the Cauchy integral of a smooth density
with respect to the arcsine distribution on [-1,1]. In this note, Nuttall's
theorem is extended to Cauchy integrals of analytic densities on the so-called
algebraic S-contours (in the sense of Nuttall and Stahl)
In general or normal random matrix ensembles, the support of eigenvalues of
large size matrices is a planar domain (or several domains) with a sharp
boundary. This domain evolves under a change of parameters of the potential and
of the size of matrices. The boundary of the support of eigenvalues is a real
section of a complex curve. Algebro-geometrical properties of this curve encode
physical properties of random matrix ensembles. This curve can be treated as a
limit of a spectral curve which is canonically defined for models of finite
matrices. We interpret the evolution of the eigenvalue distribution as a growth
problem, and describe the growth in terms of evolution of the spectral curve.
We discuss algebro-geometrical properties of the spectral curve and describe
the wave functions (normalized characteristic polynomials) in terms of
differentials on the curve. General formulae and emergence of the spectral
curve are illustrated by three meaningful examples.Comment: 44 pages, 14 figures; contains the first part of the original file.
The second part will be submitted separatel