Theoretical and Mathematical Physics

, Volume 189, Issue 1, pp 1450–1458 | Cite as

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

  • P. G. GrinevichEmail author
  • P. M. Santini


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 \( - \int_x^\infty {dx'} \) . 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.


dispersionless partial differential equation scattering transform Cauchy problem vector field Pavlov equation nonlocality tomography with an obstacle 


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© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudny, Moscow OblastRussia
  4. 4.Dipartimento di FisicaUniversità di Roma “La Sapienza,”RomeItaly
  5. 5.Istituto Nazionale di Fisica NucleareSezione di RomaRomeItaly

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