An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

Abstract

Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v t = v x v y −∂ x −1 ∂ y [v y +v x 2 ], where the formal integral ∂ x −1 becomes the asymmetric integral − x dx . We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f (X, Y ) over a parabola in the plane (X, Y ) can be expressed in terms of the integrals of f (X, Y ) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle