13,979 research outputs found
Bogoliubov Renormalization Group and Symmetry of Solution in Mathematical Physics
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.Comment: Contribution to the proceedings of conference "RG 2000" (Taxco,
Mexico, Jan. 1999). To be published in Physics Report
Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
Transfer matrices and matrix product operators play an ubiquitous role in the
field of many body physics. This paper gives an ideosyncratic overview of
applications, exact results and computational aspects of diagonalizing transfer
matrices and matrix product operators. The results in this paper are a mixture
of classic results, presented from the point of view of tensor networks, and of
new results. Topics discussed are exact solutions of transfer matrices in
equilibrium and non-equilibrium statistical physics, tensor network states,
matrix product operator algebras, and numerical matrix product state methods
for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit
A multiple exp-function method for nonlinear differential equations and its application
A multiple exp-function method to exact multiple wave solutions of nonlinear
partial differential equations is proposed. The method is oriented towards ease
of use and capability of computer algebra systems, and provides a direct and
systematical solution procedure which generalizes Hirota's perturbation scheme.
With help of Maple, an application of the approach to the dimensional
potential-Yu-Toda-Sasa-Fukuyama equation yields exact explicit 1-wave and
2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton
type solutions. Two cases with specific values of the involved parameters are
plotted for each of 2-wave and 3-wave solutions.Comment: 12 pages, 16 figure
Higher charge calorons with non-trivial holonomy
The full ADHM-Nahm formalism is employed to find exact higher charge caloron
solutions with non-trivial holonomy, extended beyond the axially symmetric
solutions found earlier. Particularly interesting is the case where the
constituent monopoles, that make up these solutions, are not necessarily
well-separated. This is worked out in detail for charge 2. We resolve the
structure of the extended core, which was previously localized only through the
singularity structure of the zero-mode density in the far field limit. We also
show that this singularity structure agrees exactly with the abelian charge
distribution as seen through the abelian component of the gauge field. As a
by-product zero-mode densities for charge 2 magnetic monopoles are found.Comment: 25 pages, 4 figures in 15 parts; small corrections - version to be
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