Related papers: Schwarzian derivative, Painlev\'e XXV-Ermakov equa…

The paper studies a non-linear transformation between Brownian martingales, which is given by the inverse of the pricing operator in the mathematical finance terminology. Subsequently, the solvability of systems of equations corresponding…

We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The…

In this paper, the exact solutions of certain non-linear differential equations defined on a fractal subset of the real line are presented. Particular attention is paid to the Riccati-type fractal differential equation, for which a…

The dynamics of nonlinear reaction-diffusion systems is dominated by the onset of patterns and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized…

Using the Chiellini condition for integrability we derive explicit solutions for a generalized system of Riccati equations $\ddot{x}+\alpha x^{2n+1}\dot{x}+x^{4n+3}=0$ by reduction to the first-order Abel equation assuming the parameter…

In 1991, one of the authors showed the existence of quadratic transformations between the Painleve' VI equations with local monodromy differences $(1/2,a,b,\pm 1/2)$ and $(a,a,b,b)$. In the present paper we give concise forms of these…

In this paper, we study the possible second order Lax operators for all the possible (1+1)-dimensional models with Schwarz variants and some special types of high dimensional models. It is shown that for every (1+1)-dimensional model and…

In this paper, we study a Malmquist-Yosida type theorem for Schwarzian differential equations \begin{equation}\label{1} S(f,z)^{m} = R(z,f) = \frac{P(z,f)}{Q(z,f)},\tag{+} \end{equation} where $m \in \mathbb{N}^{+}$, $P(z,f)$ and $Q(z,f)$…

Based on the works by Kajiwara, Noumi and Yamada, we propose a canonically quantized version of the rational Weyl group representation which originally arose as "symmetries" or the B\"acklund transformations in Painlev\'{e} equations. We…

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schr\"odinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the…

A canonical variable coefficient nonlinear Schr\"{o}dinger equation with a four dimensional symmetry group containing $\SL(2,\mathbb{R})$ group as a subgroup is considered. This typical invariance is then used to transform by a symmetry…

Recurrence coefficients of semi-classical orthogonal polynomials (orthogonal polynomials related to a weight function $w$ such that $w'/w$ is a rational function) are shown to be solutions of non linear differential equations with respect…

This work introduces a methodology to solve ordinary differential equations using the Schur decomposition of the linear representation of the differential equation. This is done by first transforming the system into an upper triangular…

We investigate the Manakov model or, more generally, the vector nonlinear Schr\"odinger equation on the half-line. Using a B\"acklund transformation method, two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet…

A generalization of the Einstein equation is considered for complex line elements. Several second order semilinear partial differential equations are derived from it as semilinear field equations in uniform and isotropic spaces. The…

We propose integrable discretizations of derivative nonlinear Schroedinger (DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS systems…

We discuss an alternative approach to Fr\'echet derivatives on Banach spaces inspired by a characterisation of derivatives due to Carath\'eodory. The approach allows us to reduce many questions of differentiability to a question of…

Let $k$ be a differential field having an algebraically closed field of constants, $E$ be a strongly normal extension of $k$, and $k^0$ be the algebraic closure of $k$ in $E.$ We prove for any intermediate differential field $k\subset…

We study reductions of the Korteweg--de Vries equation corresponding to stationary equations for symmetries from the noncommutative subalgebra. An equivalent system of $n$ second-order equations is obtained, which reduces to the Painlev\'e…

We consider fractional differential equations of order $\alpha \in (0,1)$ for functions of one independent variable $t\in (0,\infty)$ with the Riemann-Liouville and Caputo-Dzhrbashyan fractional derivatives. A precise estimate for the order…