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I want to get the differential solution numerically. I used wolfram alpha. here is the link.

I tried this code on the Mathematica file.

sols4 = NDSolve[{x''[
       r] + (1/r) x'[r] - (0.5/r^2) Sin[2 x[r]] + (2/r) Sin[x[r]]^2 - 
      0.2 Sin[x[r]] - Sin[2 x[r]] == 0, x[0.00001] == Pi, 
    x'[0.00001] == -1.1024724270140055}, x[r], {r, 0.00001, 100}, 
   Method -> "BDF"];
Plot[Evaluate[x[r] /. sols4], {r, 0.00001, 100}]

It makes me able to get the solution. But I also need a parametric plot as mentioned in wolfram alpha. How I can have that plot? Additionally, as I am using a shooting method and I need a solution to be zero at infinity. Can anyone tell me how I can avoid oscillation at large distances?

Thanks.

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  • $\begingroup$ ParametricPlot[{x[r] /. First@sols4, D[x[r] /. First@sols4, r]} // Evaluate, {r, 0.00001, 100}] $\endgroup$ – march Sep 12 at 17:03
  • $\begingroup$ Thank you very much. It worked. Do you have any idea that how I can get rid of oscillation at r>20? I need solution to be zero as like it is before r=20. $\endgroup$ – physicsu83 Sep 12 at 17:12
  • $\begingroup$ I'm sorry but I don't understand your question. The solution is the solution. Are you talking about modifying your model so that it does that or defining a new function that is the solution above until $r=20$ and then zero afterward? (And what do you mean by zero? Do you mean it's at the origin after that or that the vertical component is zero?) $\endgroup$ – march Sep 12 at 17:35
  • $\begingroup$ Actually, my differential equation is a boundary value problem. With x[0.0001]=Pi and x[infinity]=0. So I used the shooting method and converted it into an initial value problem with x[0.0001]=Pi and x'[0.0001]=-(some guessed value). But my gussed value gives oscillation at large distance. But I need x[infinity]=0 not oscillation. $\endgroup$ – physicsu83 Sep 12 at 17:42
  • $\begingroup$ You are changing the question. Please open a new question that includes both the code above and the boundary condition at large r that you are trying to match. My guess is that you are trying to match a separatrix at large `r', which typically is difficult but possible numerically. $\endgroup$ – bbgodfrey Sep 12 at 18:27
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sols4 = NDSolveValue[{x''[
       r] + (1/r) x'[r] - (0.5/r^2) Sin[2 x[r]] + (2/r) Sin[x[r]]^2 - 
      0.2 Sin[x[r]] - Sin[2 x[r]] == 0, x[0.00001] == Pi, 
    x'[0.00001] == -1.1024724270140055}, x, {r, 0.00001, 100}, 
   Method -> "BDF"];

Plot[sols4[r], {r, 0.00001, 100}]

enter image description here

For a parametric plot use ParametricPlot

ParametricPlot[
 {sols4[r], sols4'[r]}, {r, 0.00001, 100}]

enter image description here

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