%0 Journal Article %D 2014 %T On an isomonodromy deformation equation without the Painlevé property %A Boris Dubrovin %A Andrey Kapaev %X 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. %I Maik Nauka-Interperiodica Publishing %G en %U http://hdl.handle.net/1963/6466 %1 6410 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-08T11:29:56Z No. of bitstreams: 1 1301.7211v2.pdf: 415681 bytes, checksum: f606a3b5df4b201ae37a9d6fa1b79016 (MD5) %R 10.1134/S1061920814010026 %0 Report %D 2013 %T On the tritronquée solutions of P$_I^2$ %A Tamara Grava %A Andrey Kapaev %A Christian Klein %X

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.

%I SISSA %G en %1 7282 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Tamara Grava (grava@sissa.it) on 2014-01-14T18:35:53Z No. of bitstreams: 1 tritronquee_coeff.pdf: 753719 bytes, checksum: 812d268b2abe25ccbcc69eb40ff75f1f (MD5)