Related papers: Darboux integrable discrete equations possessing a…
This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations,…
We study the integrability of mappings obtained as reductions of the discrete Korteweg-de Vries (KdV) equation and of two copies of the discrete potential Korteweg-de Vries equation (pKdV). We show that the mappings corresponding to the…
In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation.…
We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we…
We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by…
The relation between the Darboux transformation and the solutions of the full Kostant Toda lattice is analyzed. The discrete Korteweg de Vries equation is used to obtain such solutions and the main result of [1] is extended to the case of…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
We review Darboux-Crum transformation of Heun's differential equation. By rewriting an integral transformation of Heun's differential equation into a form of elliptic functions, we see that the integral representation is a generalization of…
We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on…
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of…
Using the q-version of the Darboux transform we obtain the general solution of q-difference Riccati equation from a special one by the action of one-parameter group. This allows us to construct the solutions for the latge class of…
M.F. Singer [Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673--688] proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential…
We point out the relevance of the Differential Galois Theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete…
We develop a linear theory of discrete complex analysis on general quad-graphs, continuing and extending previous work of Duffin, Mercat, Kenyon, Chelkak and Smirnov on discrete complex analysis on rhombic quad-graphs. Our approach based on…
The autonomous Duffing oscillator, and its van der Pol modification, are known to admit time-dependent first integrals for specific values of parameters. This corresponds to the existence of Darboux polynomials, and in fact more can be…
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of…
Discrete gradient methods are a class of numerical integrators producing solutions with exact preservation of first integrals of ordinary differential equations. In this paper, we apply order theory combined with the symmetrized Itoh--Abe…
Darboux transformation is developed to systematically find variable separation solutions for the Nizhnik-Novikov-Veselov equation. Starting from a seed solution with some arbitrary functions, the once Darboux transformation yields the…
Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricetd to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational…
We consider discrete nets in Grassmannians $\mathbb{G}^d_r$ which generalize Q-nets (maps $\mathbb{Z}^N\to\mathbb{P}^d$ with planar elementary quadrilaterals) and Darboux nets ($\mathbb{P}^d$-valued maps defined on the edges of…