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This article is cited in 4 scientific papers (total in 4 papers)
An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects
P. G. Grinevichabc, P. M. Santinide a Landau Institute for Theoretical Physics,
Chernogolovka, Russia
b Lomonosov Moscow State University, Moscow,
Russia
c Moscow Institute of Physics and Technology, Dolgoprudny,
Moscow Oblast, Russia
d Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Roma, Italy
e Dipartimento di Fisica, Università di Roma "La Sapienza", Roma, Italy
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 $v_t=v_xv_y-\partial^{-1}_x\,\partial_y[v_y+v^2_x]$, 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.
Citation:
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”, TMF, 189:1 (2016), 59–68; Theoret. and Math. Phys., 189:1 (2016), 1450–1458
Linking options:
https://www.mathnet.ru/eng/tmf9098https://doi.org/10.4213/tmf9098 https://www.mathnet.ru/eng/tmf/v189/i1/p59
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