Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fourth Painlev\'e Equation and $PT$-Symmetric Hamiltonians   This paper is an addendum to earlier papers \cite{R1,R2} in which it was
shown that the unstable separatrix solutions for Painlev\'e I and II are
determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix
solutions of the fourth Painlev\'e transcendent are studied numerically and
analytically. For a fixed initial value, say $y(0)=1$, a discrete set of
initial slopes $y'(0)=b_n$ give rise to separatrix solutions. Similarly, for a
fixed initial slope, say $y'(0)=0$, a discrete set of initial values $y(0)=c_n$
give rise to separatrix solutions. For Painlev\'e IV the large-$n$ asymptotic
behavior of $b_n$ is $b_n\sim B_{\rm IV}n^{3/4}$ and that of $c_n$ is $c_n\sim
C_{\rm IV} n^{1/2}$. The constants $B_{\rm IV}$ and $C_{\rm IV}$ are determined
both numerically and analytically. The analytical values of these constants are
found by reducing the nonlinear Painlev\'e IV equation to the linear eigenvalue
equation for the sextic $PT$-symmetric Hamiltonian $H=\frac{1}{2}
p^2+\frac{1}{8} x^6$.

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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph