The Isomonodromic Deformation Method in the Theory of Painlevé Equations [electronic resource] / by Alexander R. Its, Victor Yu. Novokshenov.

Monodromy data for the systems of linear ordinary differential equations with rational coefficients -- Isomonodromic deformations of systems of linear ordinary differential equations with rational coefficients -- Isomonodromic deformations of systems (1.9) and (1.26) and painlevé equations of II and III types -- Inverse problem of the monodromy theory for the systems (1.9) and (1.26). Asymptotic analysis of integral equations of the inverse problem -- Asymptotic solution to a direct problem of the monodromy theory for the system (1.9) -- Asymptotic solution to a direct problem of the monodromy theory for the system (1.26) -- The manifold of solutions of painlevé II equation decreasing as ? ? ??. Parametrization of their asymptotics through the monodromy data. Ablowitz-segur connection formulae for real-valued solutions decreasing exponentially as ? ? + ? -- The manifold of solutions to painlevé III equation. The connection formulae for the asymptotics of real-valued solutions to the cauchy problem -- The manifold of solutions to painlevé II equation increasing as ? ? + ?. The expression of their asymptotics through the monodromy data. The connection formulae for pure imaginary solutions -- The movable poles of real-valued solutions to painlevé II equation and the eigenfunctions of anharmonic oscillator -- The movable poles of the solutions of painlevé III equation and their connection with mathifu functions -- Large-time asymptotics of the solution of the cauchy problem for MKdV equation -- The dynamics of electromagnetic impulse in a long laser amplifier -- The scaling limit in two-dimensional ising model -- Quasiclassical mode of the three-dimensional wave collapse.