Localized solutions of the Painlevé equations (Q2781648)

Localized solutions to the Painlevé equations represent a subclass of Painlevé transcendents exponentially vanishing at infinity. Here, the author considers another example of such solution arising naturally for a two-dimensional sine-Gordon equation \(\varphi_{xx}+ \varphi_{yy}= \sin(\varphi)\), admitting point-like singularities of the type \(\varphi\approx \alpha\log{1\over r}+ \beta+ O(1)\), \(r^2= x^2+ y^2\to 0\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00015].