Two-Soliton Surface

2soliton symKuen morph 001
In the 2-parameter family of 2-soliton surfaces lies this symmetric deformation of the Kuen Surface

The Two-Soliton Surface surface is the surface of Gaussian curvature K = -1 that corresponds to the following solution of the Sine-Gordon equation:

q(u,v) := 4 * arctan( B * (exp(A2(u, v)) - exp(A1(u, v))) / (1 + exp(A1(u, v) + A2(u, v))));


B := (cos(bb * π) - cos(aa * π)) / (cos((bb - aa) * π) - 1);

A1 := cosec(aa * π) * u - cotan(aa * π) * v;

A2 := cosec(bb * π) * u - cotan(bb * π) * v;
2 soliton leave symmetr 001
2-soliton surface with varying parameter, starting from the Kuen Surface.
2 soliton equal params 001
The soliton solution formula gives 0/0 when the two parameters are equal, but a limit exists. This animation is close to the limit 1-parameter solution.

Let us summarize, how the more complicated pseudospherical surfaces are constructed. If one parametrizes a surface of Gauss curvature K = -1 in special (so called “asymptotic”) coordinates, then the angle q(x,t) between the parameter lines determines the so called first and second fundamental forms of the surface. Once these forms are given, the surface can be obtained by solving (numerically if necessary) a pair of ODEs for the parameter lines. The fundamental theorem of surface theory says: a necessary and sufficient condition for this construction is, that the two fundamental forms satisfy the Gauss- and Codazzi- integrability conditions. In the case of our special asymptotic coordinates these integrability conditions simplify enormously to:

“The angle function q(x,t) must satisfy the Sine-Gordon equation (SGE)”

, which is: d/dt d/dx q(x,t) = sin(q(x,t))

A second function Q(x,t) is called a “Bäklund transformation with parameter s” if Q(x,t) satisfies a pair of first order equations:

Q(x,t) can be found from these two ODEs, if q(x,t) is an SGE solution and then Q(x,t) is also an SGE solution. If one starts with q(x,t) = 0 then one obtains Q(x,t) = 4 * arctan(exp(s*x + t/s)) . From Q(x,t) the Dini Surface are obtained.

An SGE marvel is that any pair s1, s2 of Bäklund parameters (with SGE solutions Q1(x,t) , Q2(x,t) allow to explicitly give another SGE solution Qq(x,t) , namely: tan( (Qq(x,t) - q(x,t))/4 ) = (s1+s2)/(s1-s2)*tan ( (Q1(x,t) - Q2(x,t))/4 ) . The 2-Soliton solutions are obtained by choosing any real pairs s = s1 , s = s2 (with s the parameter in the Bäcklund transformation pairs of ODEs above). The construction also works with complex parameters s1 = z, s2 = conj(z) .

In this case Q1 and Q2 are not real functions, but Qq turns out to be real again. If |z| = 1 , one gets the surfaces called “breathers” and for other z one gets deformations of breathers like the image above.

The interesting parts of these pseudospherical surfaces tend to be much longer in one direction (“from end to end”) than orthogonal to it.
two soliton st
two soliton st
two soliton sw
two soliton sw

Related Surface

  1. Dini Surface
  2. Breather Surface
  3. Kuen Surface
  4. Two-Soliton Surface
  5. Three-Soliton
  6. Four-Soliton Surface

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