The Isomonodromic Deformation Method in the Theory of Painleve Equations

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 painlev¿quations 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 painlev¿I 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 painlev¿II equation. The connection formulae for the asymptotics of real-valued solutions to the cauchy problem.- The manifold of solutions to painlev¿I 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 painlev¿I equation and the eigenfunctions of anharmonic oscillator.- The movable poles of the solutions of painlev¿II 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.