A numerical solution of the differential equation \(u''+ 2u'/r = u-u^ 3\) (Q1201955)

The solution of the spherically symmetric case of the problem \(\Delta u=u-u^ 3\), where \(\Delta\) is the three-dimensional Laplace operator, satisfies the equation (1) \(u''(r)+(1/r)u'(r)=u(r)-u^ 3(r)\), with the initial conditions (2) \(u(0)=\alpha\), \(u'(0)=0\). The author constructs an algorithm to find the value of \(\alpha\) for which the solution of (1), (2) approaches zero as \(r\to\infty\). It is based on a modification of the shooting method combined with the bisection of the feasible initial values. Some numerical results obtained from computer experiments are given.