As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of
integrable dispersionless multidimensional PDEs is non-local, and the proper
choice of integration constants should be the one dictated by the associated
Inverse Scattering Transform (IST). Using the recently made rigorous IST for
vector fields associated with the so-called Pavlov equation
vxtβ+vyyβ+vxβvxyββvyβvxxβ=0, we have recently esatablished that, in
the nonlocal part of its evolutionary form vtβ=vxβvyβββxβ1ββyβ[vyβ+vx2β], the formal
integral βxβ1β corresponding to the solutions of the Cauchy
problem constructed by such an IST is the asymmetric integral
ββ«xββdxβ². In this paper we show that this results could be guessed
in a simple way using a, to the best of our knowledge, novel integral geometry
lemma. Such a lemma establishes that it is possible to express the integral of
a fairly general and smooth function f(X,Y) over a parabola of the (X,Y)
plane in terms of the integrals of f(X,Y) over all straight lines non
intersecting the parabola. A similar result, in which the parabola is replaced
by the circle, is already known in the literature and finds applications in
tomography. Indeed, in a two-dimensional linear tomographic problem with a
convex opaque obstacle, only the integrals along the straight lines
non-intersecting the obstacle are known, and in the class of potentials
f(X,Y) with polynomial decay we do not have unique solvability of the inverse
problem anymore. Therefore, for the problem with an obstacle, it is natural not
to try to reconstruct the complete potential, but only some integral
characteristics like the integral over the boundary of the obstacle. Due to the
above two lemmas, this can be done, at the moment, for opaque bodies having as
boundary a parabola and a circle (an ellipse).Comment: LaTeX, 13 pages, 3 figures. arXiv admin note: substantial text
overlap with arXiv:1507.0820