Painlevé III equation
= Painlevé III equation
{c}
The third Painlevé equation is the nonlinear second-order <ordinary differential equation>
$$
w''=\frac{(w')^2}{w}-\frac{w'}z
+\frac{\alpha w^2+\beta}{z}
+\gamma w^3+\frac\delta w.
$$
The substitution $w=e^{iF}$ in the <Scaling symmetry of the sine-Gordon equation>[scaling reduction of the sine-Gordon equation] gives the parameter choice $(\alpha,\beta,\gamma,\delta)=(1/2,-1/2,0,0)$.