01015nas 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-pi200778nas a2200109   4500008004300000245004900043210004800092520045100140100002300591700001800614856003600632       2006                        en_Ud  00aThomae type formulae for singular Z_N curves0 aThomae type formulae for singular ZN curves3 aWe give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szego kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki\\\'s proof [31], obtained for non-singular ZN curves.1 aEnolski, Victor Z.1 aGrava, Tamara  uhttp://hdl.handle.net/1963/2125