Example: Constant astigmatism equation: Difference between revisions
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==Zero curvature representation== | ==Zero curvature representation== | ||
Let | |||
<math> | <math> | ||
A = \left(\begin{array}{cc} | A = \left(\begin{array}{cc} | ||
\displaystyle \frac{1}{8} \frac{\lambda^2 + 1) z_x}{\lambda z} + \frac{1}{8} \frac{(-\lambda^2 + 1) z_y}{\lambda} & | \displaystyle \frac{1}{8} \frac{(\lambda^2 + 1) z_x}{\lambda z} + \frac{1}{8} \frac{(-\lambda^2 + 1) z_y}{\lambda} & | ||
\displaystyle \frac{1}{4} \frac{(\lambda + 1)^2 \sqrt{z}}{\lambda} \\ \\ | \displaystyle \frac{1}{4} \frac{(\lambda + 1)^2 \sqrt{z}}{\lambda} \\ \\ | ||
\displaystyle \frac{1}{4} \frac{(\lambda - 1)^2 \sqrt{z}}{\lambda} & | \displaystyle \frac{1}{4} \frac{(\lambda - 1)^2 \sqrt{z}}{\lambda} & | ||
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\displaystyle \frac{1}{4} \frac{-\lambda^2 + 1}{\lambda \sqrt{z}} & | \displaystyle \frac{1}{4} \frac{-\lambda^2 + 1}{\lambda \sqrt{z}} & | ||
\displaystyle -\frac{1}{8} \frac{(-\lambda^2 + 1) z_x}{\lambda z^2} - \frac{1}{8} \frac{(\lambda^2 + 1) z_y}{\lambda z} | \displaystyle -\frac{1}{8} \frac{(-\lambda^2 + 1) z_x}{\lambda z^2} - \frac{1}{8} \frac{(\lambda^2 + 1) z_y}{\lambda z} | ||
\end{array}\right) | \end{array}\right). | ||
</math> | </math> | ||
[B-M] | Then <math>D_y - D_x + [A,B] = 0.</math> [B-M] | ||
Revision as of 12:42, 6 December 2013
Equation
[B-M]
Zero curvature representation
Let
Then [B-M]
References
[B-M] H. Baran and M. Marvan, On integrability of Weingarten surfaces: a forgotten class, J. Phys. A: Math. Theor 42 (2009) 404007.