Breather surface

In differential equations, a breather surface is a mathematical surface relating to breathers.

Breather surface when ${\displaystyle a={\frac {2}{5}}{\text{ and }}-14\leq u<14}$${\displaystyle {\text{ and }}-37.4\leq v<37.4}$

Parameterization

${\displaystyle {\begin{aligned}x&{}=-u+{\frac {2\left(1-a^{2}\right)\cosh(au)\sinh(au)}{a\left(\left(1-a^{2}\right)\cosh ^{2}(au)+a^{2}\,\sin ^{2}\left({\sqrt {1-a^{2}}}v\right)\right)}}\\\\y&{}={\frac {2{\sqrt {1-a^{2}}}\cosh(au)\left(-{\sqrt {1-a^{2}}}\cos(v)\cos \left({\sqrt {1-a^{2}}}v\right)-\sin(v)\sin \left({\sqrt {1-a^{2}}}v\right)\right)}{a\left(\left(1-a^{2}\right)\cosh ^{2}(au)+a^{2}\,\sin ^{2}\left({\sqrt {1-a^{2}}}v\right)\right)}}\\\\z&{}={\frac {2{\sqrt {1-a^{2}}}\cosh(au)\left(-{\sqrt {1-a^{2}}}\sin(v)\cos \left({\sqrt {1-a^{2}}}v\right)+\cos(v)\sin \left({\sqrt {1-a^{2}}}v\right)\right)}{a\left(\left(1-a^{2}\right)\cosh ^{2}(au)+a^{2}\,\sin ^{2}\left({\sqrt {1-a^{2}}}v\right)\right)}}\end{aligned}}}$

where 0 < a < 1.

References

• Xah Lee Web - Surface Gallery