arXiv subjects
Publications and source records attributed to A. A. Kapaev.
Using the Riemann-Hilbert approach, we explicitly construct the asymptotic $Ψ$-function corresponding to the solution $y\sim\pm\sqrt{-x/2}$ as $|x|\to\infty$ to the second Painlevé equation $y_{xx}=2y^3+xy-α$. We precisely describe the exponentially small jump in the dominant solution and the coefficient asymptotics in its power-like expansion.
The asymptotics of the generic second Painleve transcendent in the complex domain is found and justified via the direct asymptotic analysis of the associated Riemann-Hilbert problem based on the Deift-Zhou nonlinear steepest descent method. The asymptotics is proved of the Boutroux type, i.e. it is expressed in terms of the elliptic functions. Explicit connection formulae between the asymptotic phases in the different sectors are obtained as well.
Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlevé equation $y_{xx}=2y^3+xy-α$. The precise description of the exponentially small jump in the dominant solution approaching $α/x$ as $|x|\to\infty$ is given. For the asymptotic power expansion of the dominant solution, the coefficient asymptotics is found.
The isomonodromy deformation equation for a 2x2 matrix linear ODE with a large parameter can be locally reduced to a (hyper)elliptic equation. To globalize this result, we apply the isomonodromy deformation method and obtain the modulation equations for the asymptotic algebraic curve. The method is applied to the degenerate solution of the Painleve first equation.
Nonlinear Schrödinger equation (with the Schwarzian initial data) is important in nonlinear optics, Bose condensation and in the theory of strongly correlated electrons. The asymptotic solutions in the region $x/t={\cal O}(1)$, $t\to\infty$, can be represented as a double series in $t^{-1}$ and $\ln t$. Our current purpose is the description of the asymptotics of the coefficients of the series.