How would one go about plotting this parameterized curve using numerical resources (analitically it's too hard)?

Question

I'm working with the system of differential equations: $$\begin{align*} \left\{ \begin{array} { l l } { (u')^2 + (v')^2 = 1 } \\ {u'v'' - u''v' = -v' + u' } \end{array} \right. \end{align*}$$ Where $u, v:I \to \mathbb{R}$. Now, with a little bit of work, one can show that (below $c$ is a constant): $$u'(s) = \frac { 2 \sqrt { 2 } \operatorname { tanh } \left[ \frac { s } { \sqrt { 2 }}+c \right] - 2 \operatorname { tanh } \left[ \frac { s } { \sqrt { 2 } } +c\right ] ^ { 2 } } { 2 - 2 \sqrt { 2 } \tanh \left[ \frac { s } { \sqrt { 2 } } +c\right] + 2 \tanh \left[ \frac { s } { \sqrt { 2 } }+c \right] ^ { 2 } }$$

And if there was a nice primitive to this function, my life would be easier, since $v = \pm \int \sqrt{1-(u')^2} \ ds$. But unfortunately there isn't. My ultimate goal here is to plot the curve: $$X(s) = (\cos(u(s)),\sin(u(s)), v(s) )$$ But due to how nasty all these expressions become, it's proven to be a pretty hard task. Naturally I think the only way for me to really "see" this curve is using numerical resources, but I'm new to Mathematica (so I'm sorry if this question is too basic, I wouldn't know), so I don't know how to do that and I'd really appreciate some help.

Answer 1

This system has an analytical solution. I will indicate it, but I can not prove that it is the only one. We start with the parametrization $u'=f=\cos(p(s)),v'=g=\sin(p(s))$. The first equation is automatically satisfied, and the second equation has an analytical solution that can be used to construct the curve $X(s)$:

f = Cos[p[s]];
g = Sin[p[s]];
f' = -Sin[p[s]]*p'[s];
g' = Cos[p[s]]*p'[s];
sol = DSolve[f*g' - g*f' + g - f == 0, p, s]

The solution of equation

{{p -> Function[{s}, -2 ArcTan[
      1 - Sqrt[2] Tanh[1/2 (Sqrt[2] s + Sqrt[2] C[1])]]]}}

Using this solution, we find functions $u,v$:

U = DSolve[u'[s] == Cos[p[s]] /. sol, u, s]
V = DSolve[v'[s] == Sin[p[s]] /. sol, v, s]

The solutions of equations:

{{u -> Function[{s}, 
    C[2] + 1/
      4 (2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]] + 
        Log[Cos[2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]] + 
          Sin[2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]]]) (Cos[
         2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]] + 
        Sin[2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]]) (-2 Cosh[
          Sqrt[2] (s + C[1])] + Sqrt[2] Sinh[Sqrt[2] (s + C[1])])]}}

{{v -> Function[{s}, 
    C[2] - 1/
      4 (2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]] - 
        Log[Cos[2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]] + 
          Sin[2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]]]) (Cos[
         2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]] + 
        Sin[2 ArcTan[1 - Sqrt[2] Tanh[(s + C[1])/Sqrt[2]]]]) (-2 Cosh[
          Sqrt[2] (s + C[1])] + Sqrt[2] Sinh[Sqrt[2] (s + C[1])])]}}

The constant C[2] in solutions $u,v$ is not the same, but you can equate them, since this does not affect the shape of the curve $X(s)$:

X = Flatten[
  Simplify[{Cos[u[s]] /. U, Sin[u[s]] /. U, v[s] /. V}, 
   TimeConstraint -> 1]];
ParametricPlot3D[X /. {C[1] -> 0, C[2] -> 0}, {s, -10, 10}, 
 BoxRatios -> {1, 1, 1}, ColorFunction -> Hue]