Related papers: Which differential equations correspond to the Lin…
In this short note, we review several one-dimensional problems such as those involving linear Schroedinger equation, variable-coefficient Helmholtz equation, Zakharov-Shabat system and Kubelka-Munk equations. We show that they all can be…
We have already dealt with the problem of solving First Order Differential Equations (1ODEs) presenting elementary functions before in [1, 2]. In this present paper, we have established solid theoretical basis through a relation between the…
We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the…
We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense…
We consider a class of linear ODEs of second order with variable coefficients and construct its Lie algebra of Lie group of equivalence transformations. Further we find invariants and differential invariants of this Lie algebra and by using…
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special…
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second…
There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator…
Time-dependent Lindblad master equations have important applications in areas ranging from quantum thermodynamics to dissipative quantum computing. In this paper we outline a general method for writing down exact solutions of time-dependent…
The notions of equivalence and strict equivalence for order one differential equations are introduced. The more explicit notion of strict equivalence is applied to examples and questions concerning autonomous equations and equations having…
We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
Open-system dynamics play a key role in the experimental and theoretical study of cavity optomechanical systems. In many cases, the quantum Langevin equations have enabled excellent models for optical decoherence, yet a master-equation…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
The quantum dynamics of a damped and forced harmonic oscillator is investigated in terms of a Lindblad master equation. Elementary algebraic techniques are employed allowing for example to analyze the long time behavior, i.e. the quantum…
Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order…
Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as…
In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential…
We introduce a variational hybrid classical-quantum algorithm to simulate the Lindblad master equation and its adjoint for time-evolving Markovian open quantum systems and quantum observables. Our method is based on a direct representation…
We develop a new method for the construction of one-dimensional integrable Lindblad systems, which describe quantum many body models in contact with a Markovian environment. We find several new models with interesting features, such as…