Related papers: From fully-nonlinear to semilinear evolution equat…
We prove that any evolution equation admitting a potential symmetry can always be reduced to another evolution equation such that the potential symmetry in question maps into the group of its contact symmetries. Based on this fact is out…
We prove, for a wide class of semilinear elliptic differential and pseudodifferential equations in $\R^d$, that the solutions which are sufficiently regular and have a certain decay at infinity extend to holomorphic functions in sectors of…
We propose a new method for solution of the integrability problem for evolutionary differential-difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is…
Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, $r$-matrices and integrals of motion in involution are explicitly proposed for…
We introduce an integrable Hamiltonian system which Lax deforms the Dirac operator D=d+d* on a finite simple graph or compact Riemannian manifold. We show that the nonlinear isospectral deformation always leads to an expansion of the…
We consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. In this setting, we prove the existence of solutions…
This short note provides positive answers to two conjectures of Camacho, Khudoyberdiyev, and Omirov on the classification of complete evolution algebras. Our approach is based on analysing the solution set of a generic non-linear polynomial…
All non-equivalent integrable evolution equations of third order of the form $u_t=D_x\frac{\delta H}{\delta u}$ are found.
In the present paper we consider a general family of two dimensional wave equations which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated…
Direct and inverse initial-boundary problems on a bounded interval for systems of quasilinear evolution equations with general nonlinearities are considered. In the case of inverse problems conditions of integral overdetermination are…
Chapter 4 of my book [1] is dedicated to Lie-B\"acklund group analysis of various types of differential equations. The Russian edition of this book had an Addendum (pages 262-266) containing a summary of new results (obtained by the end of…
In this work we develop some fifth-order integrable coupled systems of weight $0$ and $1$ which possess seventh-order symmetry. We establish four new systems, where in some cases, related recursion operator and bi-Hamiltonian formulations…
An integrable hierarchies connected with linear stationary Schr\"odinger equation with energy dependent potentials (in general case) are considered. Galilei-like and scaling invariance transformations are constructed. A symmetry method is…
We consider differential-difference equations that determine the continuous symmetries of discrete equations on the triangular lattice. It is shown that a certain combination of continuous flows can be represented as a scalar evolution…
In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for…
The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth…
We analyze three different high-order nonlinear Schr\"{o}dinger equation (HONLSE) models that have been used in the literature to describe the evolution of slowly modulated gravity waves on the surface of ideal finite-depth fluid. We…
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms…
We construct integrable discrete nonautonomous quad-equations as B\"acklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by…
We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are…