dorsal/arxiv
View SchemaNew exact solutions for the discrete fourth Painlev\'e equation
| Authors | Andrew P. Bassom, Peter A. Clarkson |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9409002 |
| URL | https://arxiv.org/abs/solv-int/9409002 |
| DOI | 10.1016/0375-9601(94)91294-7 |
Abstract
In this paper we derive a number of exact solutions of the discrete equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})= {-2z_nx_n^3+(\eta-3\delta^{-2}-z_n^2)x_n^2+\mu^2\over (x_n+z_n+\gamma)(x_n+z_n-\gamma)},\eqno(1)$$ where $z_n=n\delta$ and $\eta$, $\delta$, $\mu$ and $\gamma$ are constants. In an appropriate limit (1) reduces to the fourth \p\ (PIV) equation $${\d^2w\over\d z^2} = {1\over2w}\left({\d w\over\d z}\right)^2+\tfr32w^3 + 4zw^2 + 2(z^2-\alpha)w +{\beta\over w},\eqno(2)$$ where $\alpha$ and $\beta$ are constants and (1) is commonly referred to as the discretised fourth Painlev\'e equation. A suitable factorisation of (1) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable $z_n$. Limits of these solutions yield rational solutions of PIV (2). It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can be obtained by analysis of (1).
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"abstract": "In this paper we derive a number of exact solutions of the discrete equation\n$$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})=\n{-2z_nx_n^3+(\\eta-3\\delta^{-2}-z_n^2)x_n^2+\\mu^2\\over\n(x_n+z_n+\\gamma)(x_n+z_n-\\gamma)},\\eqno(1)$$ where $z_n=n\\delta$ and $\\eta$,\n$\\delta$, $\\mu$ and $\\gamma$ are constants. In an appropriate limit (1) reduces\nto the fourth \\p\\ (PIV) equation $${\\d^2w\\over\\d z^2} = {1\\over2w}\\left({\\d\nw\\over\\d z}\\right)^2+\\tfr32w^3 + 4zw^2 + 2(z^2-\\alpha)w +{\\beta\\over\nw},\\eqno(2)$$ where $\\alpha$ and $\\beta$ are constants and (1) is commonly\nreferred to as the discretised fourth Painlev\\\u0027e equation. A suitable\nfactorisation of (1) facilitates the identification of a number of solutions\nwhich take the form of ratios of two polynomials in the variable $z_n$. Limits\nof these solutions yield rational solutions of PIV (2). It is also known that\nthere exist exact solutions of PIV (2) that are expressible in terms of the\ncomplementary error function and in this article we show that a discrete\nanalogue of this function can be obtained by analysis of (1).",
"arxiv_id": "solv-int/9409002",
"authors": [
"Andrew P. Bassom",
"Peter A. Clarkson"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1016/0375-9601(94)91294-7",
"title": "New exact solutions for the discrete fourth Painlev\\\u0027e equation",
"url": "https://arxiv.org/abs/solv-int/9409002"
},
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