The asymptotic behavior of solutions of the sine-Gordon equation with singularities zero
The author deals with the case of a radially symmetric equation of the form \(u_{xx}+{1\over x} u_x+ \sin u= 0\) and studies the asymptotic behaviour of its solution. It turns out that the solutions with singularity of the form \[ u\sim \alpha\log{1\over x}+ \beta+ O(1),\quad x^2= y^2+ t^2\to 0,\quad \alpha,\beta\in \mathbb{R} \] at zero have an asymptotic limit at infinity: \[ u(x)= 2\pi k+ \gamma\sqrt{{\pi\over 2x}} e^{-x}[1+ O(x^{-1})],\quad x\to\infty, \] where \(k\) is an integer. Moreover, the corresponding modulation equation and phase shift are stated.
scientific article; zbMATH DE number 1531333