The nonlinear Sine-Gordon partial differential equation

$$ 2\psi^{\prime \prime} = \sin\, 2 \psi /a^2 \tag1 $$

in its asymptotic form belongs to hyperbolic geometry, is a hyperbolic pde, results in Chebychev net on the Pseudosphere along asymptotic lines. It relates to the Breather and Soliton solutions all of them have only a negative Gauss curvature $K$.


From more rudimentary consideration earlier, took a cue from my intuition about surface area of spherical shell segments (axi-symmetric integration/ computation) and applied it onto pseudo-sphere. These involve switching between constant axial / radial length difference factor respectively. The segment areas have a formula of same form:

$$ 2 \pi r_{max} \Delta z \rightarrow2 \pi r_{max} \Delta r \tag2 $$


Next, in Clairaut’s Law of geodesics we have $ r\cdot \sin \psi$ as an invariant, entirely radius $r$ based ( $r$ radius, $\psi$ angle to meridian).

So now I attempted to make formulation $dz$ centric instead of $dr$ centric or chose $\phi$ instead of its complement angle by a swap. On making this and related changes (i.e., $\sin \psi$ proportional to $z$ rather than $r$ etc.,) and after numerical integration to find surface shape I obtained Flat corrugated disks which obey Sine-Gordon differential equation.

It is surprising (to me) that it consists of both positive and negative Gauss curvature patches. After verification I am posting some representative images here appearing as a new result.

Obeying SGE

The meridians are the same as for $K<0$ case but rotated lying on same side of cusp locus plane and have curly parentheses shaped meridians on either side for two chosen constants. The Gauss curvature is seen to vary wildly, even if the surface in its $z$ coordinate is bounded.

Now, in what way are the modified lines shown in red color asymptotic ? The asymptotic lines are seen even when $K>0,\, ! $


The Sine-Gordon satisfied in (1) is satisfied here also.

If the meridional tangent makes $\phi$ to axis we have:

$$ 2\phi^{\prime \prime} + \sin\, 2 \phi /r_o^2 =0\, \tag3 $$

$$ a\,\sin\phi = \sqrt{(p-z) (q+z)}, \tag4 $$

where $(a,p,q,r_o)$ are constants. So we have both types of grooves with cuspidal edges bounded between two planes in $z$ direction and periodic along radius. First type is a bumpy spiral and the second one is a helix along an intersecting tube.


Six important conclusions can be drawn for this corrugated radial plate model:

1) The meridian profile is same as those for $K<0$ pseudospheres but do not run parallel to $z$ axis of symmetry but rotated to be parallel to $r$ axis. (They are reflected about lines parallel to $r=z$).

2) The Gauss curvature sign varies from positive to negative and its magnitude is not constant.

3) The lines shown cannot be asymptotic. $ K>0$ parts of half toroid can have no such asymptotes.

4) A radial plane touches cuspidal edges at points having zero slope and $\psi= (2k-1)\pi/2, \phi=0. $ which is the cusp locus.

5) There is no Chebychev net, but a new corrugated/crimped net.

6) In spite of such stark differences with the axial model Sine-Gordon Equation is still obeyed to be among surfaces satisfying SGE.


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