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## Homework Statement

I have been trying to solve the following nonlinear ordinary differential equation:

##-\Phi''-\frac{3}{r}\Phi'+\Phi-\frac{3}{2}\Phi^{2}+\frac{\alpha}{2}\Phi^{3}=0##

with boundary conditions ##\Phi'(0)=0,\Phi(\infty)=0.##

## Homework Equations

My solution is supposed to reproduce the following plots:

## The Attempt at a Solution

Now, to produce the plots given above, I wrote the following Mathematica code:

\[Alpha] = 0.99;

\[CapitalPhi]lower = 0;

\[CapitalPhi]upper = 5;

For[counter = 0, counter <= 198, counter++,

\[CapitalPhi]0 = (\[CapitalPhi]lower + \[CapitalPhi]upper)/2;

r0 = 0.00001;

\[CapitalPhi]r0 = \[CapitalPhi]0 + (1/16) (r0^2) (2 (\[CapitalPhi]0) - 3 (\[CapitalPhi]0^2) + \[Alpha] (\[CapitalPhi]0^3));

\[CapitalPhi]pr0 = (1/8) (r0) (2 (\[CapitalPhi]0) - 3 (\[CapitalPhi]0^2) + \[Alpha] (\[CapitalPhi]0^3));

diffeq = {-\[CapitalPhi]''[r] - (3/r) \[CapitalPhi]'[r] + \[CapitalPhi][r] - (3/2) (\[CapitalPhi][r]^2) + (\[Alpha]/2) (\[CapitalPhi][r]^3) == 0, \[CapitalPhi][r0] == \[CapitalPhi]r0, \[CapitalPhi]'[r0] == \[CapitalPhi]pr0};

sol = NDSolve[diffeq, \[CapitalPhi], {r, r0, 200}, Method -> "ExplicitRungeKutta"];

\[CapitalPhi]test = \[CapitalPhi][200] /. sol[[1]];

\[CapitalPhi]upper = If[(\[CapitalPhi]test < 0) || (\[CapitalPhi]test >

1.2), \[CapitalPhi]0, \[CapitalPhi]upper];

\[CapitalPhi]lower = If[(\[CapitalPhi]test < 1.2) && (\[CapitalPhi]test > 0), \[CapitalPhi]0, \[CapitalPhi]lower];

]

Plot[Evaluate[{\[CapitalPhi][r]} /. sol[[1]]], {r, 0, 200}, PlotRange -> All, PlotStyle -> Automatic]

In the code, I used Taylor expansion at ##r=0## due to the ##-\frac{3}{r}\Phi'## term. Moreover, I used shooting method and continually bisected an initial interval from ##\Phi_{\text{upper}}=5## to ##\Phi_{\text{lower}}=0## to obtain more and more precise values of ##\Phi(0)##.

With the code above, I was able to produce the plots for ##\alpha = 0.50, 0.90, 0.95,0.96,0.97##. For example, my plot for ##\alpha = 0.50## is as follows:

However, my plot for ##\alpha = 0.99## does not converge to the required plot:

Can you suggest how I might tackle this problem for ##\alpha = 0.99##? Also, is there an explanation for the plots shooting upwards and oscillating after a prolonged asymptotic trend towards the positive ##r##-axis?

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