Breather (Parametric)

Parametric Breather Surface Parametric Equations:

r := 1 - b^2;
w := Sqrt[r];
denom := b*((w*Cosh[b*u])^2 + (b*Sin[w*v])^2)
breather = {-u + (2*r*Cosh[b*u]*Sinh[b*u])/
        denom, (2*w*Cosh[b*u]*(-(w*Cos[v]*Cos[w*v]) - Sin[v]*Sin[w*v]))/
      denom, (2*w*Cosh[b*u]*(-(w*Sin[v]*Cos[w*v]) + Cos[v]*Sin[w*v]))/denom}

ParametricPlot3D[
  Evaluate[breather /. b -> 0.4], {u, -13.2, 13.2}, {v, -37.4, 37.4}, 
  PlotRange -> All, PlotPoints -> {60, 150}]

u controls how far the tip goes. v controls the girth. b varies from 0 to 1

The above image has boundary {u,-13.2,13.2} and {v,-37.4,37.4}, with b=0.4.