I’m trying to solve these two coupled 2nd order differential equations:

with the following boundary conditions:

where $r_{*}$ is the value of the $r$ at midpoint. I was trying to solve these equations using shooting method and for $l=3$ and $t=4$, I Succeeded:

```
m = 1/2 (Tanh(v(x)/(1/3)) + 1);
rv3 = ParametricNDSolve({r(x)^4/(Guess)^2 - r(x)^2 -
2 r'(x) v'(x) + (r(x)^2 - m) v'(x)^2 == 0,
r(x)^2 - r(x)^2 v'(x)^2 - r(x) v''(x) + 2 r'(x) v'(x) == 0,
r(-1.5) == 10, v(-1.5) == 4, v'(-1.5) == guess}, {r, v}, {x, -1.5,
1.5}, {guess, Guess}, MaxSteps -> Infinity)
Manipulate(
Plot(Evaluate(r(guess, Guess)(t) /. rv3), {t, -1.5, 1.5}), {{Guess,
1.1039}, 1.1, 1.2}, {{guess, -9.051}, -10, -9})
Manipulate(
Plot(Evaluate(v(guess, Guess)(t) /. rv3), {t, -1.5, 1.5}), {{Guess,
1.1039}, 1.1, 1.2}, {{guess, -9.051}, -10, -9})
```

but for length intervals greater $l$ than 3, for example 6, I’m having problem finding the right value for guessing. Also we have symmetry along the $x$-axis at midpoint and the derivates of $r$ and $v$ with respect to $x$ are both zero $r’=v’=0$ but for lengths greater than 3, I keep getting solutions that doesn’t respect the symmetry and doesn’t have those derivatives zero at midpoint.

I’m trying to get some results similar to these:

I could replicate the $l=3$ one but for the rest, especially $l=6$ and greater I’m having problems finding the right values, because I’m getting solutions that are not correct. For example something like this for the $v-x$ plot:

where I’ve chose `r(-6)==100`

.

Can anyone point me in the right direction so I can find the values for the equations? any help would be appreciated.