@article {2013,
title = {On an isomonodromy deformation equation without the Painlev{\'e} property},
number = {Russian Journal of Mathematical Physics},
year = {2014},
note = {34 pages, 8 figures, references added},
publisher = {Maik Nauka-Interperiodica Publishing},
abstract = {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.},
doi = {10.1134/S1061920814010026},
url = {http://hdl.handle.net/1963/6466},
author = {Boris Dubrovin and Andrey Kapaev}
}
@article {10979,
title = {On the tritronqu{\'e}e solutions of P$_I^2$},
year = {2013},
institution = {SISSA},
abstract = {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.

},
author = {Tamara Grava and Andrey Kapaev and Christian Klein}
}