Timelike W-Surfaces in Minkowski 3-Space ℝ13 and the Sinh-Gordon Equation

Let M and M∗ be two timelike surfaces in Minkowski 3-space ℝ13. If there exists a spacelike (timelike) Darboux line congruence between each point of M and M∗, then the surfaces are timelike Weingarten surfaces. It turns out their Tschebyscheff angles are solutions of the Sinh-Gordon equation, and the surfaces are related to each other by Backlund’s transformation. Finally, a method to construct new timelike Weingarten surface has been found.