Related papers: From fully-nonlinear to semilinear evolution equat…
A general form of the fifth-order nonlinear evolution equation is considered. Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type…
The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…
We analyze nonlinear Schr\"odinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.
We present a list of (1+1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution…
Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative B\"acklund chart, generalizing results in [Fuchssteiner B., Carillo S.,…
Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two equations admitting simple Lie algebras of dimension three. Next, we prove that…
The paper is devoted to give a complete classification of five-dimension nilpotent evolution algebras over an algebraically closed field. We obtained a list of 27 isolated non-isomorphic nilpotent evolution algebras and 2 families of…
We expand our group classification of quasilinear evolution equations (Acta Appl.Math., v.69, 2001) to the case of general evolution equation in one spatial variable. This enables obtaining several new classes of evolution equations with…
We introduce a method of approximate nonclassical Lie-B\"acklund symmetries for partial differential equations with a small parameter and discuss applications of this method to finding of approximate solutions both integrable and…
The problem of linearization for third order evolution equations is considered. Criteria for testing equations for linearity are presented. A class of linearizable equations depending on arbitrary functions is obtained by requiring presence…
Linearization of coupled second order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper we describe a simple and straightforward method to…
The family of fifth order nonlinear evolution equations is studied. Some traveling wave elliptic solutions are found. The classification of these exact solutions is given.
We study the symmetry reduction of nonlinear evolution and wave type differential equations by using operators of non-point symmetry. In our approach we use both operators of classical and conditional symmetry. It appears that the…
We study the symmetry reduction of nonlinear partial differential equations which are used for describing diffusion processes in nonhomogeneous medium. We find ansatzes reducing partial differential equations to systems of ordinary…
We propose a general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations. This allows us to give the general integrable evolution corresponding to an arbitrary time and space dependence of the…
We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of…
In the present paper the non-Noether symmetries of the Toda model, nonlinear Schodinger equation and Korteweg-de Vries equations (KdV and mKdV) are discussed. It appears that these symmetries yield the complete sets of conservation laws in…
We prove that, for $m\ge 7$, scalar evolution equations of the form $u_t=F(x,t,u,...,u_m)$ which admit a nontrivial conserved density of order $m+1$ are linear in $u_m$. The existence of such conserved densities is a necesary condition for…
The survey provides classification results for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on necessary integrability conditions following from the existence of the…
Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a…