00957nas a2200121   4500008004100000245007500041210006900116260004100185520053400226100002000760700001900780856003600799       2014                        en  d00aOn an isomonodromy deformation equation without the Painlevé property0 aisomonodromy deformation equation without the Painlevé property  bMaik Nauka-Interperiodica Publishing3 aWe 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.1 aDubrovin, Boris1 aKapaev, Andrey  uhttp://hdl.handle.net/1963/646601015nas a2200133   4500008004100000245004500041210003400086260001000120520062000130100001800750700001900768700002100787856007300808       2013                        en  d00aOn the tritronquée solutions of P$_I^2$0 atritronquée solutions of PI2  bSISSA3 a<p>For equation P$_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.</p>1 aGrava, Tamara1 aKapaev, Andrey1 aKlein, Christian  uhttps://www.math.sissa.it/publication/tritronqu%C3%A9e-solutions-pi2