Related papers: A Generalized Cole-Hopf Transformation for Nonline…
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…
A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new…
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…
We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by…
Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy.…
We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the ('Floquet-Lyapunov')…
Equations of motion corresponding to the H\'{e}non - Heiles system are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential…
In earlier papers the author studied some classes of equations with Carlitz derivatives for $\mathbb F_q$-linear functions, which are the natural function field counterparts of linear ordinary differential equations. Here we consider…
In this paper we present a direct formula for the solution of the general second order linear ordinary differential equation as our main result such that the parameters required for the formula are determined using another differential…
We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then,…
This work investigates a class of moving boundary problems related to a nonlinear evolution equation featuring an exponential source term. We establish a connection to Stefan-type problems, for different boundary conditions at the fixed…
Bagderina \cite{Bagderina2013} solved the equivalence problem for a family of scalar second-order ordinary differential equations (ODEs), with cubic nonlinearity in the first-order derivative, via point transformations. However, the…
We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \,…
We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a…
We present a new technique for constructing solutions of quasilinear systems of first-order partial differential equations, in particular inhomogeneous ones. A generalization of the Riemann invariants method to the case of inhomogeneous…
We systematically introduce the idea of applying differential operator method to find a particular solution of an ordinary nonhomogeneous linear differential equation with constant coefficients when the nonhomogeneous term is a polynomial…
We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations,…
The technique of Darboux transformation is applied to nonlocal partner of two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear…
Here we study the abstract nonlinear differential equation of second order that in special case is the equation of the type of equation of traffic flow. We prove the solvability theorem for the posed problem under the appropriate conditions…
We generalize the nonlinear one-dimensional equation for a fluid layer surface to any geometry and we introduce a new infinite order differential equation for its traveling solitary waves solutions. This equation can be written as a…