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
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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.
@article{TMF_2016_189_1_a4,
author = {P. G. Grinevich and P. M. Santini},
title = {An~integral geometry lemma~and its applications: {The~nonlocality} of {the~Pavlov} equation and a~tomographic problem with opaque parabolic objects},
journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
pages = {59--68},
publisher = {mathdoc},
volume = {189},
number = {1},
year = {2016},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a4/}
}
TY - JOUR AU - P. G. Grinevich AU - P. M. Santini TI - An~integral geometry lemma~and its applications: The~nonlocality of the~Pavlov equation and a~tomographic problem with opaque parabolic objects JO - Teoretičeskaâ i matematičeskaâ fizika PY - 2016 SP - 59 EP - 68 VL - 189 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a4/ LA - ru ID - TMF_2016_189_1_a4 ER -
%0 Journal Article %A P. G. Grinevich %A P. M. Santini %T An~integral geometry lemma~and its applications: The~nonlocality of the~Pavlov equation and a~tomographic problem with opaque parabolic objects %J Teoretičeskaâ i matematičeskaâ fizika %D 2016 %P 59-68 %V 189 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/TMF_2016_189_1_a4/ %G ru %F TMF_2016_189_1_a4
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/