2
$\begingroup$

I want to solve the following differential equation

f''[z] + (1/z)*f'[z] - f[z] + f[z]^3 == 0

subject to the boundary conditions

$$f^{\prime}(0)=0\qquad\lim_{z\to\infty}f(z)=0$$ The solution for R subject to these boundary conditions is known as Townes soliton. I have followed what is shown in the example solved here: https://mathematica.stackexchange.com/a/156362/73726.

What I have tried so far is the following:

sol = NDSolveValue[{f''[z] + (1/z)*f'[z] - f[z] + f[z]^3 == 0, f[5] == 0, f'[0] == 0}, f, {z, 0, 5}, Method -> { "Shooting", "StartingInitialConditions" ->{f[5] == 100, f'[0] == 0}}];Plot[sol[z], {z, 0, 5}, AxesLabel -> {z, f}]

But it is retuning the following errors

NDSolve`Shooting::ndcinit: Initial conditions should be specified at a single point.

NDSolveValue::dsvar: 0.00010214285714285715` cannot be used as a variable.

General::stop: Further output of NDSolveValue::dsvar will be suppressed during this calculation.

Improve this question
$\endgroup$
2
  • $\begingroup$ Please include your expression in MMA form and the code you have tried so far. $\endgroup$
    – MarcoB
    Commented Apr 9, 2021 at 18:59
  • $\begingroup$ @MarcoB I have just updated it. Hope it is adequated now. $\endgroup$
    – HeitorGalacian
    Commented Apr 9, 2021 at 19:13

1 Answer 1

Reset to default
1
$\begingroup$

Maybe this helps:

 $Version
 (*"12.2.0 for Microsoft Windows (64-bit) (December 12, 2020)"*)

 sol = With[{e = 10^-30}, 
 NDSolveValue[{f''[z] + (1/z)*f'[z] - f[z] + f[z]^3 == 0, 
 f[200] == 0, f'[e] == 0}, f, {z, e, 200}, 
 Method -> {"Shooting", 
 "StartingInitialConditions" -> {f[e] == -21, f'[e] == 0}}, 
 MaxSteps -> 10^6, WorkingPrecision -> 25]]; Plot[
 sol[z], {z, 0, 200}, AxesLabel -> {z, f}, PlotRange -> All]

enter image description here

Improve this answer
$\endgroup$
1
  • $\begingroup$ The command of Maple 2021 evalf(dsolve({diff(f(z), z, z) + diff(f(z), z)/z - f(z) + f(z)^3 = 0, f(0) = 1/3, D(f)(0) = 0}, f(z), series, order = 20)) produces $f(z)= 0.3333333333+ 0.07407407407 z^{2}+ 0.003086419753 z^{4}- 0.00009525986892 z^{6}- 0.00001448743840 z^{8}- 5.587696941\times10^{-7} z^{10}+ 1.259010856\times10^{-8} z^{12}+ 2.608548074\times10^{-9} z^{14}+ 1.185138978\times10^{-10} z^{16}- 1.313655828\times10^{-12} z^{18}+\mathrm{O}(z^{20}) $. Since the coefficients rapidly decrease, it looks similar to a trunskated power expansion of a certain entire function. $\endgroup$
    – user64494
    Commented Apr 10, 2021 at 12:32

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.