AN INTEGRAL GEOMETRY LEMMA AND ITS APPLICATIONS: THE NONLOCALITY OF THE PAVLOV EQUATION AND A TOMOGRAPHIC PROBLEM WITH OPAQUE PARABOLIC OBJECTS

作者：Grinevich P G^*; Santini P M

来源：Theoretical and Mathematical Physics, 2016, 189(1): 1450-1458.

DOI：10.1134/S0040577916100056

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

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)-- partial derivative(-1)(x)partial derivative(y)[v(y)+ v(x)(2)], where the formal integral partial derivative(-1)(x) x becomes the asymmetric integral - integral(infinity)(x) dx(1) 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.

• 出版日期2016-10