dorsal/arxiv
View SchemaString equation--2. Physical solution
| Authors | P. G. Grinevich, S. P. Novikov |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9501002 |
| URL | https://arxiv.org/abs/solv-int/9501002 |
| Journal | Algebra and Analysis v. 6 No. 3 (1994) 118-140 (in russian); english translation -- St. Petersburg Math. J. v. 6 No. 3 (1995) 553-574 |
Abstract
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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"abstract": "This paper is a continuation of the paper by S.P.Novikov in Funct.Anal.Appl.,\nv.24(1990), No 4, pp 196-206. String equation is by definition the equation\n$[L,A]=1$ for the coefficients of two linear ordinary differential operators\n$L$ and $A$. For the ``double scaling limit\u0027\u0027 of the matrix model we always\nhave $L=-\\partial_x^2+u(x)$, $A$ is some differential operator of the odd order\n$2k+1$. In the first nontrivial case $k=1$ we have the Painelev\\\u0027e-1 (P-1)\nequation. Only special real ``separatrix\u0027\u0027 solutions of P-1 are important in\nthe quantum field theory. By the conjecture of Novikov these ``physical\u0027\u0027\nsolutions, which are analytically exceptional probably have much stronger\nsymmetry then the other solutions but it is not proved until now. Two\nasymptotic methods were developed in the previous paper -- nonlinear\nsemiclassics (or the Bogolubov-Whitham averaging method) and the linear\nsemiclassics for the ``Isomonodromic\u0027\u0027 method. The nonlinear semiclassics gives\na good approximation for the general (``non-physical\u0027\u0027) solutions of P-1 but\nfails in the ``physical\u0027\u0027 case. In our paper the linear semiclasics for the\n``physical\u0027\u0027 solutions of the P-1 equations is studied. In particular\nconnection between the semiclassics on Riemann surfaces and Hamiltonian\nfoliations on these surfaces is established.",
"arxiv_id": "solv-int/9501002",
"authors": [
"P. G. Grinevich",
"S. P. Novikov"
],
"categories": [
"solv-int",
"hep-th",
"nlin.SI"
],
"journal_ref": "Algebra and Analysis v. 6 No. 3 (1994) 118-140 (in russian);\n english translation -- St. Petersburg Math. J. v. 6 No. 3 (1995) 553-574",
"title": "String equation--2. Physical solution",
"url": "https://arxiv.org/abs/solv-int/9501002"
},
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