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Papers by Dimitris Tsoubelis
Comptes Rendus Mathematique, 2009
The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms... more The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann-Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space-time variables. For an appropriate set of initial and boundary data, we derive the k-dependent "spectral functions" which guarantee the uniqueness of Riemann-Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I.
Journal of Computational and Applied Mathematics, 2009
Complete symmetry groups enable one to characterise fully a given differential equation. By consi... more Complete symmetry groups enable one to characterise fully a given differential equation. By considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of Bateman, Monge-Ampère and Born-Infeld as special cases. We develop a symbolic algorithm to decrease the complexity of the calculations involved.
Journal of Physics: Conference Series, 2007
The Lie point symmetries of the Einstein vacuum equations corresponding to the Bondi form of the ... more The Lie point symmetries of the Einstein vacuum equations corresponding to the Bondi form of the line element are presented. Using these symmetries, we study reductions of the field equations, which might lead to new asymptotically flat solutions, representing gravitational waves emitted by an isolated source.
Mathematical Physics Analysis and Geometry, 1998
The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular re... more The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 ≤ x < y ≤ 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possesses a Lax pair formulation. Using the simultaneous spectral analysis of this Lax pair we study the above Goursat problem as well as its linearized version. It is shown that the linear problem reduces to a scalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problem reduces to a 2 × 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.
Astrophysical Journal, 1988
This paper considers a spherical body consisting of a fluid with heat flow which radiates in its ... more This paper considers a spherical body consisting of a fluid with heat flow which radiates in its exterior a null fluid described by the outgoing Vaidya's metric. A Friedmann-like exact solution of the interior Einstein field equations is given. It is proved that this solution, matched with the outgoing Vaidya matric, represents a physically reasonble collapsing model which, when the heat flow is switched off, reduces to the well-known collapsing model with dust. The proposed model has the remarkable property that even if the heat flow is small, the horizon will never be formed because, before this happens, the collapsing body will be destroyed by opposite gradients of pressure.
Astrophysical Journal, 1988
A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional ... more A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional reduction of the anti-self-dual Yang-Mills equations, is presented. It represents a generalization of the Ernst-Weyl equation of General Relativity related to colliding neutrino and gravitational waves, as well as of the fourth order equation of Schwarzian type related to the KdV hierarchies, which was introduced by Nijhoff, Hone, and Joshi recently. An auto-Bäcklund transformation of the new system is constructed, leading to a superposition principle remarkably similar to the one connecting four solutions of the KdV equation. At the level of the Ernst-Weyl equation, this Bäcklund transformation and the associated superposition principle yield directly a generalization of the single and double Harrison transformations of the Ernst equation, respectively. The very method of construction also allows for revealing, in an essentially algorithmic fashion, other integrability features of the main subsystems, such as their reduction to the Painlevé transcendents.
General Relativity and Gravitation, 1987
The motion of test particles in polar orbit about the source of the Kerr field of gravity is stud... more The motion of test particles in polar orbit about the source of the Kerr field of gravity is studied, using Carter's first integrals for timelike geodesies in the Kerr space-time. Expressions giving the angular coordinates of such particles as functions of the radial one are derived, both for the case of a rotating black hole as well as for that of a naked singularity.
Physics Letters A, 1986
A gyroscope following a closed polar orbit in the Kerr spacetime is considered. An exact expressi... more A gyroscope following a closed polar orbit in the Kerr spacetime is considered. An exact expression is derived giving the shift of the gyroscope's orientation per revolution in terms of the mass and angular momentum parameters of the Kerr metric and the orbit's coordinate radius.
General Relativity and Gravitation, 1989
A new three-parameter class of solutions to the Einstein vacuum equations is presented which repr... more A new three-parameter class of solutions to the Einstein vacuum equations is presented which represents the collision of a pair of gravitational plane waves. Depending on the choice of the parameters, one of the colliding waves has a smooth or unbounded wavefront, or it is a shock, or impulsive, or shock accompanied by an impulsive wave, while the second is any of the above types. A subfamily of the solutions develops no curvature singularity in the interaction region formed by the colliding waves.
General Relativity and Gravitation, 1988
The gravitational field along the symmetry axis of the Kerr spacetime is examined. The equations ... more The gravitational field along the symmetry axis of the Kerr spacetime is examined. The equations of parallel transport along this axis are solved for the timelike geodesics case, and the corresponding tidal tensor is constructed.
Physical Review Letters, 1988
Comptes Rendus Mathematique, 2009
The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms... more The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann-Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space-time variables. For an appropriate set of initial and boundary data, we derive the k-dependent "spectral functions" which guarantee the uniqueness of Riemann-Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given. To cite this article: I.
Journal of Computational and Applied Mathematics, 2009
Complete symmetry groups enable one to characterise fully a given differential equation. By consi... more Complete symmetry groups enable one to characterise fully a given differential equation. By considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of Bateman, Monge-Ampère and Born-Infeld as special cases. We develop a symbolic algorithm to decrease the complexity of the calculations involved.
Journal of Physics: Conference Series, 2007
The Lie point symmetries of the Einstein vacuum equations corresponding to the Bondi form of the ... more The Lie point symmetries of the Einstein vacuum equations corresponding to the Bondi form of the line element are presented. Using these symmetries, we study reductions of the field equations, which might lead to new asymptotically flat solutions, representing gravitational waves emitted by an isolated source.
Mathematical Physics Analysis and Geometry, 1998
The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular re... more The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 ≤ x < y ≤ 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possesses a Lax pair formulation. Using the simultaneous spectral analysis of this Lax pair we study the above Goursat problem as well as its linearized version. It is shown that the linear problem reduces to a scalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problem reduces to a 2 × 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.
Astrophysical Journal, 1988
This paper considers a spherical body consisting of a fluid with heat flow which radiates in its ... more This paper considers a spherical body consisting of a fluid with heat flow which radiates in its exterior a null fluid described by the outgoing Vaidya's metric. A Friedmann-like exact solution of the interior Einstein field equations is given. It is proved that this solution, matched with the outgoing Vaidya matric, represents a physically reasonble collapsing model which, when the heat flow is switched off, reduces to the well-known collapsing model with dust. The proposed model has the remarkable property that even if the heat flow is small, the horizon will never be formed because, before this happens, the collapsing body will be destroyed by opposite gradients of pressure.
Astrophysical Journal, 1988
A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional ... more A new system of integrable nonlinear equations of hyperbolic type, obtained by a two-dimensional reduction of the anti-self-dual Yang-Mills equations, is presented. It represents a generalization of the Ernst-Weyl equation of General Relativity related to colliding neutrino and gravitational waves, as well as of the fourth order equation of Schwarzian type related to the KdV hierarchies, which was introduced by Nijhoff, Hone, and Joshi recently. An auto-Bäcklund transformation of the new system is constructed, leading to a superposition principle remarkably similar to the one connecting four solutions of the KdV equation. At the level of the Ernst-Weyl equation, this Bäcklund transformation and the associated superposition principle yield directly a generalization of the single and double Harrison transformations of the Ernst equation, respectively. The very method of construction also allows for revealing, in an essentially algorithmic fashion, other integrability features of the main subsystems, such as their reduction to the Painlevé transcendents.
General Relativity and Gravitation, 1987
The motion of test particles in polar orbit about the source of the Kerr field of gravity is stud... more The motion of test particles in polar orbit about the source of the Kerr field of gravity is studied, using Carter's first integrals for timelike geodesies in the Kerr space-time. Expressions giving the angular coordinates of such particles as functions of the radial one are derived, both for the case of a rotating black hole as well as for that of a naked singularity.
Physics Letters A, 1986
A gyroscope following a closed polar orbit in the Kerr spacetime is considered. An exact expressi... more A gyroscope following a closed polar orbit in the Kerr spacetime is considered. An exact expression is derived giving the shift of the gyroscope's orientation per revolution in terms of the mass and angular momentum parameters of the Kerr metric and the orbit's coordinate radius.
General Relativity and Gravitation, 1989
A new three-parameter class of solutions to the Einstein vacuum equations is presented which repr... more A new three-parameter class of solutions to the Einstein vacuum equations is presented which represents the collision of a pair of gravitational plane waves. Depending on the choice of the parameters, one of the colliding waves has a smooth or unbounded wavefront, or it is a shock, or impulsive, or shock accompanied by an impulsive wave, while the second is any of the above types. A subfamily of the solutions develops no curvature singularity in the interaction region formed by the colliding waves.
General Relativity and Gravitation, 1988
The gravitational field along the symmetry axis of the Kerr spacetime is examined. The equations ... more The gravitational field along the symmetry axis of the Kerr spacetime is examined. The equations of parallel transport along this axis are solved for the timelike geodesics case, and the corresponding tidal tensor is constructed.
Physical Review Letters, 1988