TY - JOUR
T1 - On an isomonodromy deformation equation without the PainlevĂ© property
Y1 - 2014
A1 - Boris Dubrovin
A1 - Andrey Kapaev
AB - We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.
PB - Maik Nauka-Interperiodica Publishing
UR - http://hdl.handle.net/1963/6466
N1 - 34 pages, 8 figures, references added
U1 - 6410
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - RPRT
T1 - On the tritronquĂ©e solutions of P$_I^2$
Y1 - 2013
A1 - Tamara Grava
A1 - Andrey Kapaev
A1 - Christian Klein
AB - For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and $t=o(x^{2/3})$. Using this result, we identify the most degenerate solutions $u^{(m)}(x,t)$, $\hat u^{(m)}(x,t)$, $m=0,\dots,6$, called {\em tritronqu\'ee}, describe the quasi-linear Stokes phenomenon and find the large $n$ asymptotics of the coefficients in a formal expansion of these solutions. We supplement our findings by a numerical study of the tritronqu\'ee solutions.

PB - SISSA
U1 - 7282
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -