On the tritronquée solutions of P${}_I^2$

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 $u^{(m)}(x,t)$, ${\hat{u}}^{(m)}(x,t)$, $m=0,...,6$, called {\em tritronquée}, 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 tritronquée solutions.

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Cited by in corpus (2)

• Numerical Approach to Painlevé Transcendents on Unbounded Domains
• Tronquée Solutions of the Third and Fourth Painlevé Equations