31 research outputs found
Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
We represent the partition function of the Generalized Kontsevich Model (GKM)
in the form of a Toda lattice -function and discuss various implications
of non-vanishing "negative"- and "zero"-time variables: the appear to modify
the original GKM action by negative-power and logarithmic contributions
respectively. It is shown that so deformed -function satisfies the same
string equation as the original one. In the case of quadratic potential GKM
turns out to describe {\it forced} Toda chain hierarchy and, thus, corresponds
to a {\it discrete} matrix model, with the role of the matrix size played by
the zero-time (at integer positive points). This relation allows one to discuss
the double-scaling continuum limit entirely in terms of GKM, essentially
in terms of {\it finite}-fold integrals.Comment: 46
Stability of Spatial Optical Solitons
We present a brief overview of the basic concepts of the soliton stability
theory and discuss some characteristic examples of the instability-induced
soliton dynamics, in application to spatial optical solitons described by the
NLS-type nonlinear models and their generalizations. In particular, we
demonstrate that the soliton internal modes are responsible for the appearance
of the soliton instability, and outline an analytical approach based on a
multi-scale asymptotic technique that allows to analyze the soliton dynamics
near the marginal stability point. We also discuss some results of the rigorous
linear stability analysis of fundamental solitary waves and nonlinear impurity
modes. Finally, we demonstrate that multi-hump vector solitary waves may become
stable in some nonlinear models, and discuss the examples of stable
(1+1)-dimensional composite solitons and (2+1)-dimensional dipole-mode solitons
in a model of two incoherently interacting optical beams.Comment: 34 pages, 9 figures; to be published in: "Spatial Optical Solitons",
Eds. W. Torruellas and S. Trillo (Springer, New York
SOLITARY WAVE SOLUTIONS FOR SPACE-TIME FRACTIONAL COUPLED INTEGRABLE DISPERSIONLESS SYSTEM VIA GENERALIZED KUDRYASHOV METHOD
In this article, space-time fractional coupled integrable dispersionless system is considered, and we use fractional derivative in the sense of modified Riemann-Liouville. The fractional system has reduced to an ordinary differential system by fractional transformation and the generalized Kudryashov method is applied to obtain exact solutions. We also testify performance as well as precision of the applied method by means of numerical tests for obtaining solutions. The obtained results have been graphically presented to show the properties of the solutions
Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
The Kazakov-Migdal model, if considered as a functional of external fields,
can be always represented as an expansion over characters of group. The
integration over "matter fields" can be interpreted as going over the {\it
model} (the space of all highest weight representations) of . In the case
of compact unitary groups the integrals should be substituted by {\it discrete}
sums over weight lattice. The version of the model is the Generalized
Kontsevich integral, which in the above-mentioned unitary (discrete) situation
coincides with partition function of the Yang-Mills theory with the target
space of genus and holes. This particular quantity is always a
bilinear combination of characters and appears to be a Toda-lattice
-function. (This is generalization of the classical statement that
individual characters are always singular KP -functions.) The
corresponding element of the Universal Grassmannian is very simple and somewhat
similar to the one, arising in investigations of the string models.
However, under certain circumstances the formal sum over representations should
be evaluated by steepest descent method and this procedure leads to some more
complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the
simple "character phase" deserves further investigation.Comment: 29 pages, UUITP-10/93, FIAN/TD-07/93, ITEP-M4/9
Analysis of optical solitons solutions of two nonlinear models using analytical technique
Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.</p
Identification problem for damped sine-Gordon equation with point sources
AbstractWe establish the existence and uniqueness of solutions for sine-Gordon equations in a multidimensional setting. The equations contain a point-like source. Furthermore, the continuity and the Gâteaux differentiability of the solution map is established. An identification problem for parameters governing the equations is set, and is shown to have a solution. The objective function is proved to be Fréchet differentiable with respect to the parameters. An expression for the Fréchet derivative in terms of the solutions of the direct and the adjoint systems is presented. A criterion for optimal parameters is formulated as a bang-bang control principle. An application of these results to the one-dimensional sine-Gordon equation is considered