Rational Solutions and Bäcklund Transformations for the Third Painlevé Equation
Rational Solutions and Bäcklund Transformations for the Third Painlevé Equation
复制标题
第三 Painlevé 方程的有理解和 Bäcklund 变换
DOI：
10.1007/978-94-011-2082-1_33
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发表时间：
1993
期刊：
影响因子：
1.7
通讯作者：
P. Clarkson
中科院分区：
文献类型：
--
作者：
A. Milne;P. Clarkson
In this paper we discuss rational solutions, one-parameter families of solutions expressible in terms of Bessel functions for the third Painleve equation (PIII)
$$ \frac{{{d^2}y}}{{d{x^2}}} = \frac{1}{y}{\left( {\frac{{dy}}{{dx}}} \right)^2} - \frac{1}{x}\frac{{dx}}{{dy}} + \frac{{\alpha {y^2} + \beta }}{x} + \gamma {y^3} + \frac{\delta }{y} $$
(PIII)
where α, β, γ and δ are arbitrary constants. PIII is interesting since it arises in many physical applications and also as a similarity reduction of several soliton equations. Using the Backlund transformations for PIII we construct hierarchies of exact solutions.
In this paper we discuss rational solutions, one-parameter families of solutions expressible in terms of Bessel functions for the third Painleve equation (PIII)
$$ \frac{{{d^2}y}}{{d{x^2}}} = \frac{1}{y}{\left( {\frac{{dy}}{{dx}}} \right)^2} - \frac{1}{x}\frac{{dx}}{{dy}} + \frac{{\alpha {y^2} + \beta }}{x} + \gamma {y^3} + \frac{\delta }{y} $$
(PIII)
where α, β, γ and δ are arbitrary constants. PIII is interesting since it arises in many physical applications and also as a similarity reduction of several soliton equations. Using the Backlund transformations for PIII we construct hierarchies of exact solutions.