Analytic Solutions of Matrix Riccati Equations with Analytic Coefficients

For matrix Riccati equations of platoon-type systems and of systems arising from PDEs, assuming the coefficients are analytic or rational functions in a suitable domain, analyticity of the stabilizing solution is proved under various hypotheses. General results on analytic behavior of stabilizing solutions are developed as well

Analytic structure and power-series expansion of the Jost function for the two-dimensional problem

For a two-dimensional quantum mechanical problem, we obtain a generalized power-series expansion of the S-matrix that can be done near an arbitrary point on the Riemann surface of the energy, similarly to the standard effective range expansion. In order to do this, we consider the Jost-function and analytically factorize its momentum dependence that causes the Jost function to be a multi-valued function. The remaining single-valued function of the energy is then expanded in the power-series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain a semi-analytic expression for the Jost-function (and therefore for the S-matrix) near an arbitrary point on the Riemann surface and use it, for example, to locate the spectral points (bound and resonant states) as the S-matrix poles. The method is applied to a model simlar to those used in the theory of quantum dots.Comment: 42 pages, 9 figures, submitted to J.Phys.

Perturbation analysis of Markov modulated fluid models

We consider perturbations of positive recurrent Markov modulated fluid models. In addition to the infinitesimal generator of the phases, we also perturb the rate matrix, and analyze the effect of those perturbations on the matrix of first return probabilities to the initial level. Our main contribution is the construction of a substitute for the matrix of first return probabilities, which enables us to analyze the effect of the perturbation under consideration

The model equation of soliton theory

We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most interesting result is the simple equation for the generating function of the hierarchy which defines the dynamics for the negative times and also has applications to the second order spectral problems. A rather general theory of integrable 1+1-dimensional equations can be developed by study of polynomial solutions of this equation under condition of regularity of the corresponding potentials.Comment: 17