Radial-symmetric solution of the cosh-Laplace equation and the distribution of its singularities (Q1267419)

A particular case of the third Painlevé equation in the form of cosh-Laplace equation \[ u_{\alpha\alpha}+ x^{-1}u_x= \text{cosh }u\tag{1} \] is considered. This choice is motivated by geometric applications. The basic tool is the isomonodromic deformation method. An asymptotic distribution of poles at infinity is obtained for the real-valued solution with arbitrary initial conditions at the origin. One of the most important results is formulated as the theorem: The general complex-valued solution to the Painlevé equation (1) in the sector \(-\pi/4< \arg x<\pi/4\) has the following asymptotics as \(| x|\to\infty\) \[ u(x/\mu,s)= -2\ln cn(0,5\sqrt{\mu+ \mu^{-1}} (\alpha- x_0)/(\mu+ \mu^{-1})^{1/2}+ \ln\mu+ O(1), \] with \(\mu(\varphi)= A(\varphi)+ \sqrt{A(\varphi)^2+1}\), \(-\pi/4< \varphi< \pi/4\), \(r\), \(s\) are arbitrary real-valued constants, \(| r|< 2\). The modulus function \(A\) is defined by some integral equation, the phase shift \(x_0= x_0(A,r,s)\) is given by an explicit formula.