excuse my ignorance...(for, this is really a difficult subject :)What exactly is a soliton? 

I was referring to another image in your new surface gallery, a related topic. Please ask Prof. R. Palais about solitons.
Xah
Also I do not get the Breather the way it is seen in the link given below, the ridges are rounded like a surface of revolution, K >0.

What may be the error?

Narasimham

<< RealTime3D`; Clear[u, v]; um = 13.2; vm = 37.4; b = 0.4 ; r =
  1 - b^2  ; w = Sqrt[r]; denom = b*((w*Cosh[b*u])^2 + (b*Sin[w*v])^2);
breather = { ( r*Sinh [2*b*u])/denom  - u,
      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[breather, {u, -um, um}, {v, -vm, vm}, PlotPoints -> {43, 23},
   AspectRatio -> Automatic];
<< Default3D`;

Breather breathtaking indeed !!

Allow me a comment about related  Three soliton surface with K= -1. One
sees K > 0 areas in the peripheral regions of the surface. I had sent a
messsage before also, to Prof. Richard Palais. I doubt if it would give
a flat 3D Plot.

Best Regards.

xah lee <xah@...> wrote:
just made an update to the breather surface, with graphing calc file &
Mathematica file.

http://www.xahlee.org/surface/breather_p/breather_p.html

  Xah

  ☄