Numerically solve 2nd order differential equations:

Question

I'm trying to solve these two coupled 2nd order differential equations, but I'm getting nowhere. Mathematica can't seem to solve these no matter what I do. The equations are:

The $r_{*}$ is the value of the $r$ at midpoint and I have the mentioned boundary value conditions:

and also I have the following symmetry along the $x$ axis where at the midpoint I have $r'(x)=v(x)=0$. I tried to solve these equations with NDsolve for the following boundary conditions but I had no luck:

m = 1/2 (Tanh[v[x]/(1/3)] + 1);
NDSolve[{r[x]^4/1.1^2 - 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] == 100000, v[1] == 4, v[2] == 4}, {r, v}, {x, 1, 2}, 
MaxSteps -> Infinity]

I have set $t=0$ and used 10000 instead of $\infty$. but unfortunately, I get the following error mesages:

Power::infy: Infinite expression 1/0. encountered.
Infinity::indet: Indeterminate expression 0. ComplexInfinity encountered.
General::stop: Further output of Power::infy will be suppressed during this calculation.
NDSolve::ndnum: Encountered non-numerical value for a derivative at x == 1.`

I would very much appreciate any help

Answer 1

This can be solved with the ShootingMethod To give you an idea of how this works, one needs to find the boundary condition that satisfies v[4]==1. So, this is an illustration of guessing:

rv = ParametricNDSolveValue[{r[x]^4/(11/10)^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] == 1, v[1] == 4, v'[1] == guess}, {r, v}, {x, 1, 5}, guess, 
  MaxSteps -> Infinity]

Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];

Manipulate[
 DynamicModule[{traj, t0, t1},
  traj = rv[guess];
  {{t0, t1}} = First[InterpolatingFunctionDomain /@ traj];
  ParametricPlot[
   Through[traj[t]], {t, t0, t1}, AspectRatio -> 1
   ]
  ],
 {{guess, 1}, -10, 10}
 ]

So, you want to get close (there may be multiple solutions as illustrated in the link) and then use the Shooting method as described there.