Rational Solutions and Bäcklund Transformations for the Third Painlevé Equation

摘要: In this paper we discuss rational solutions, one-parameter families of solutions expressible in terms 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 many physical applications also as a similarity reduction several soliton equations. Using Backlund transformations construct hierarchies exact solutions.