Related papers: A Generalized Cole-Hopf Transformation for Nonline…
We study the extension of integrable equations which possess the Lax representations to noncommutative spaces. We construct various noncommutative Lax equations by the Lax-pair generating technique and the Sato theory. The Sato theory has…
A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations,…
Miura transform is known as the transformation between Korteweg de-Vries equation and modified Korteweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type…
We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of…
We study a fourth-order derivative scalar field configuration in a fixed Lifshitz background. Using an auxiliary field we rewrite the equations of motion as two coupled second order equations. We specialize to the limit that the mass of the…
We consider a matrix nonlinear partial differential equation that generalizes Heisenberg ferromagnet equation. This generalized Heisenberg ferromagnet equation is completely integrable with a linear bundle Lax pair related to the…
We are concerned with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions at resonance, in which the nonlinear boundary conditions perturbation is not necessarily required to…
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit…
We exhibit an alternative method for solving inhomogeneous second--order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis…
We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal…
We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method…
Invariants of nonlinear gauge transformations of a family of nonlinear Schr\"odinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.
A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of the Newton polygons corresponding to nonlinear differential equations. It allows one to express exact solutions…
In this paper we present sufficient conditions for the existence of heteroclinic or homoclinic solutions for second order coupled systems of differential equations on the real line. We point out that it is required only conditions on the…
In this paper we construct Darboux transformations for the supersymmetric Two-boson equation. Two Darboux transformations and associated B\"acklund transformations are presented. For one of them, we also obtain the corresponding the…
Point-to-point reflection holding for harmonic functions subject to the Dirichlet or Neumann conditions on an analytic curve in the plane almost always fails for solutions to more general elliptic equations. We develop a non-local,…
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…
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate…
Darboux transformation of a second-order linear differential operator is a well-known technique with many applications in mathematics and physics. We study Darboux transformation from the point of view of Markov semigroups of diffusion…