String equation--2. Physical solution

Details String equation--2. Physical solution String equation--2. Physical solution

1995-01-11

arxiv.org/abs/solv-int/9501002v3

Algebra and Analysis v. 6 No. 3 (1994) 118-140 (in russian); english translation -- St. Petersburg Math. J. v. 6 No. 3 (1995)

Physics

High Energy Physics

High Energy Physics - Theory

32 pages, LaTex, 4 pictures in separate files. Subj-class and Journal-ref added

Scientific paper

This paper is a continuation of the paper by S.P.Novikov in Funct.Anal.Appl., v.24(1990), No 4, pp 196-206. String equation is by definition the equation $[L,A]=1$ for the coefficients of two linear ordinary differential operators $L$ and $A$. For the ``double scaling limit'' of the matrix model we always have $L=-\partial_x^2+u(x)$, $A$ is some differential operator of the odd order $2k+1$. In the first nontrivial case $k=1$ we have the Painelev\'e-1 (P-1) equation. Only special real ``separatrix'' solutions of P-1 are important in the quantum field theory. By the conjecture of Novikov these ``physical'' solutions, which are analytically exceptional probably have much stronger symmetry then the other solutions but it is not proved until now. Two asymptotic methods were developed in the previous paper -- nonlinear semiclassics (or the Bogolubov-Whitham averaging method) and the linear semiclassics for the ``Isomonodromic'' method. The nonlinear semiclassics gives a good approximation for the general (``non-physical'') solutions of P-1 but fails in the ``physical'' case. In our paper the linear semiclasics for the ``physical'' solutions of the P-1 equations is studied. In particular connection between the semiclassics on Riemann surfaces and Hamiltonian foliations on these surfaces is established.

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