Parametric Breather Pseudospherical Surface

Parametric breather surfaces are known in one-to-one correspondence with the solutions of a certain non-linear wave-equation, i.e., the so-called Sine-Gordon Equation. It turns out, solutions to this equation correspond to unique pseudospherical surfaces, namely soliton. Breather surface corresponds to a time-periodic 2-soliton solution.

Parametric breather surface has the following parametric equations :

$x = -u+\frac{2\left(1-a^2\right)\cosh(au)\sinh(au)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)}$

$y = \frac{2\sqrt{1-a^2}\cosh(au)\left(-\sqrt{1-a^2}\cos(v)\cos\left(\sqrt{1-a^2}v\right)-\sin(v)\sin\left(\sqrt{1-a^2}v\right)\right)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)}$

$z = \frac{2\sqrt{1-a^2}\cosh(au)\left(-\sqrt{1-a^2}\sin(v)\cos\left(\sqrt{1-a^2}v\right)+\cos(v)\sin\left(\sqrt{1-a^2}v\right)\right)}{a\left(\left(1-a^2\right)\cosh^2(au)+a^2\,\sin^2\left(\sqrt{1-a^2}v\right)\right)}$

Where $0<a<1$ , $u$ controls how far the tip goes, and $v$ controls the girth.

When $a=0.4$ , $-13<u<13$ , and $-38<v<38$ :

With orthographic projection :

About Pseudospherical Surfaces :

Surface in $\mathbb{R}^{3}$ having constant Gaussian curvature $K= -1$ are usually called pseudospherical surfaces.

If $X : M \subset \mathbb{R}^{3}$ is a surface with Gaussian curvature $K= -1$ then it is known that there exists a local asymptotic coordinate system $(x,t)$ on $M$ such that the first and second fundamental forms are:

$dx^{2}+dt^{2}+2 \cos{q}~dx~dt$ , and $2 \sin{q}~dx~dt$ ,

where $q$ is the angle between asymptotic lines (the x-curves and t-curves). The Gauss-Codazzi equations for $M$ in these coordinates become a single equation, the sine-Gordon equation (SGE) :

$q_{xt}= \sin{q}$

The SGE is one of the model soliton equations.

References and Readings :

Chuu-Lian Terng. 2004. Lecture notes on curves and surfaces in $\mathbb{R}^{3}$ , available here.
Chuu-Lian Terng. 1990s. About Pseudospherical Surfaces, available here.
Richard S Palais. 2003. A Modern Course on Curves and Surfaces, available here.

About 3D-XplorMath :

3D-XplorMath is a Mathematical Visualization program. The older original version, written in Pascal, runs only on Macintosh computers, but there is also a newer cross-platform Java version, called 3D-XplorMath-J which is written in the Java programming language; to use it, you must have Java 5.0 or higher installed on your computer.

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Parametric Breather Pseudospherical Surface

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