Related papers: From fully-nonlinear to semilinear evolution equat…
The construction of a generalized (higher-order) nonlinear thermo-hydrodynamics, based on a nonequilibrium ensemble formalism has been presented in the preceding article. The working of such theory is illustrated in the present one. We…
We will study evolution algebras $A$ which are free modules of dimension $2$ over domains. Furthermore, we will assume that these algebras are perfect, that is $A^2=A$. We start by making some general considerations about algebras over…
Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…
We perform a classification of integrable systems of mixed scalar and vector evolution equations with respect to higher symmetries. We consider polynomial systems that are homogeneous under a suitable weighting of variables. This paper…
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
We study the general properties of spectral curves associated to doubly-periodic solutions of Korteweg-deVries, sine-Gordon, Non-linear Schr\"odinger and 1D Toda equations, and construct examples of arbitrary genus.
Several classes of systems of evolution equations with one or two vector unknowns are considered. We investigate also systems with one vector and one scalar unknown. For these classes all equations having the simplest higher symmetry are…
We consider inhomogeneous supersymmetric bilinear forms, i.e., forms that are neither even nor odd. We classify such forms up to dimension seven in the case when the restrictions of the form to the even and odd parts of the superspace are…
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no…
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or…
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for linearizability of a…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
We classify the four dimensional perfect non-simple evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.
In the present paper we study some algebraic properties of evolution algebras. Moreover, we reduce the study of evolution algebras of permutations to two special types of evolution algebras, idempotents and absolute nilpotent elements of…
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
Obstacles to integrability in perturbed evolution equations are overcome by allowing higher-order terms in the expansion of the solution to depend explicitly on time and position. With a special expansion algorithm, obstacles vanish…
We complete the Lie symmetry classification of scalar nth order, $n \geq 4$, ordinary differential equations by means of the symmetry Lie algebras they admit. It is known that there are three types of such equations depending upon the…
We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of…
Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes…
In 2004, the article "Maximal regularity for evolution equations in weighted $L_p$-spaces" by J. Pr\"{u}ss and G. Simonett has been published in Archiv der Mathematik. We provide a survey of the main results of that article and outline some…