Vladimir Kovalev - Academia.edu (original) (raw)
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Papers by Vladimir Kovalev
Cornell University - arXiv, Aug 9, 2011
The paper is devoted to the group analysis of equations of motion of two-dimensional uniformly st... more The paper is devoted to the group analysis of equations of motion of two-dimensional uniformly stratified rotating fluids used as a basic model in geophysical fluid dynamics. It is shown that the nonlinear equations in question have a remarkable property to be self-adjoint. This property is crucial for constructing conservation laws provided in the present paper. Invariant solutions are constructed using certain symmetries. The invariant solutions are used for defining internal wave beams.
Classical Lie group theory provides a universal tool for calculating symmetry groups for systems ... more Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integrodifferential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benney equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Bäcklund operators.
Lecture Notes in Physics, 2010
ABSTRACT
Lecture Notes in Physics, 2010
ABSTRACT
Lecture Notes in Physics, 2010
ABSTRACT
Lecture Notes in Physics, 2010
ABSTRACT
Theoretical and Mathematical Physics, 1999
The results from developing and applying the notions of functional self-similarity and the Bogoli... more The results from developing and applying the notions of functional self-similarity and the Bogoliubov renormalization group to boundary-value problems in mathematical physics during the last decade are reviewed. The main achievement is the regular algorithm for finding renormalization group-type symmetries using the contemporary theory of Lie groups of transformations.
Physics Reports, 2001
Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is pr... more Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution. After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS). Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given. As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium.
Journal of Physics A: Mathematical and General, 2006
Recent advances in generalizing the renorm-group algorithm for boundary value problems of mathema... more Recent advances in generalizing the renorm-group algorithm for boundary value problems of mathematical physics and the related concept of the renorm-group symmetry, previously formulated with reference to models based on differential equations, are revisited. The algorithm and symmetry are now formulated for models with nonlocal (integral) equations. Examples illustrate how the updated algorithm applies to models with nonlocal terms appearing as linear functionals of the solution.
Cornell University - arXiv, Aug 9, 2011
The paper is devoted to the group analysis of equations of motion of two-dimensional uniformly st... more The paper is devoted to the group analysis of equations of motion of two-dimensional uniformly stratified rotating fluids used as a basic model in geophysical fluid dynamics. It is shown that the nonlinear equations in question have a remarkable property to be self-adjoint. This property is crucial for constructing conservation laws provided in the present paper. Invariant solutions are constructed using certain symmetries. The invariant solutions are used for defining internal wave beams.
Classical Lie group theory provides a universal tool for calculating symmetry groups for systems ... more Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integrodifferential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benney equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Bäcklund operators.
Lecture Notes in Physics, 2010
ABSTRACT
Lecture Notes in Physics, 2010
ABSTRACT
Lecture Notes in Physics, 2010
ABSTRACT
Lecture Notes in Physics, 2010
ABSTRACT
Theoretical and Mathematical Physics, 1999
The results from developing and applying the notions of functional self-similarity and the Bogoli... more The results from developing and applying the notions of functional self-similarity and the Bogoliubov renormalization group to boundary-value problems in mathematical physics during the last decade are reviewed. The main achievement is the regular algorithm for finding renormalization group-type symmetries using the contemporary theory of Lie groups of transformations.
Physics Reports, 2001
Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is pr... more Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution. After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS). Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given. As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium.
Journal of Physics A: Mathematical and General, 2006
Recent advances in generalizing the renorm-group algorithm for boundary value problems of mathema... more Recent advances in generalizing the renorm-group algorithm for boundary value problems of mathematical physics and the related concept of the renorm-group symmetry, previously formulated with reference to models based on differential equations, are revisited. The algorithm and symmetry are now formulated for models with nonlocal (integral) equations. Examples illustrate how the updated algorithm applies to models with nonlocal terms appearing as linear functionals of the solution.