Geodesic
An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects
Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 1, pp. 59-68 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $v_t=v_xv_y-\partial^{-1}_x\,\partial_y[v_y+v^2_x]$, where the formal integral $\partial^{-1}_x$ 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.
Keywords: dispersionless partial differential equation, scattering transform, Cauchy problem, vector field, Pavlov equation, nonlocality, tomography with an obstacle. 
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P. G. Grinevich; P. M. Santini. An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects. Teoretičeskaâ i matematičeskaâ fizika, Tome 189 (2016) no. 1, pp. 59-68. http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a4/

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