No Arabic abstract
An important representation of the general-type fundamental solutions of the canonical systems corresponding to matrix string equations is established using linear similarity of a certain class of Volterra operators to the squared integration. Explicit fundamental solutions of these canonical systems are also constructed via the GBDT version of Darboux transformation. Examples and applications to dynamical canonical systems are given. Explicit solutions of the dynamical canonical systems are constructed as well. Three appendices are dedicated to the Weyl--Titchmarsh theory for canonical systems, transformation of a subclass of canonical systems into matrix string equations (and of a smaller subclass of canonical systems into matrix Schrodinger equations), and a linear similarity problem for Volterra operators.
We consider canonical systems (with $2ptimes 2p$ Hamiltonians $H(x)geq 0$), which correspond to matrix string equations. Direct and inverse problems are solved in terms of Titchmarsh--Weyl and spectral matrix functions and related $S$-nodes. Procedures for solving inverse problems are given.
We obtain high energy asymptotics of Titchmarsh-Weyl functions of the generalised canonical systems generalising in this way a seminal Gesztesy-Simon result. The matrix valued analog of the amplitude function satisfies in this case an interesting new identity. The corresponding structured operators are studied as well.
We study matrix roots with certain commutation properties and their application to the explicit construction of Darboux matrices in the framework of the GBDT version of Backlund-Darboux transformation. The approach is demonstrated on the important case of generalised canonical systems depending rationally on the spectral parameter.
In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $Nge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, which is a Gibbs distribution. In this paper we study the {em speed of convergence} towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove that the convergence is indeed exponential. The rate as measured by the decay of the $L_2$ norm can be bound in terms of the spectral gap of the Laplacian of the graph, and as measured by the decay of (relative) entropy be bound using the modified logarithmic Sobolev constant of the graph. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [CHLZ] The first one is a local inequality, while the second is a global inequality with respect to the lower bound metric from [CHLZ].
We introduce matrix coupled (local and nonlocal) dispersionless equations, construct wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and consider their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and nonlocal dispersionless equations as well.