First integrals in the discrete variational calculus

The intent of this paper is to show that first integrals of the discrete Euler equation can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analog of the classical theorem of E. Noether in the Calculus of Variations.     

A variational approach to difference equations

In this paper, we are concerned with the existence of at least three distinct solutions for nonlinear difference equations with Dirichlet boundary conditions. The proof of the main result is based on variational methods. We also provide an example in order to illustrate the main results.     

On variational methods for nonlinear difference equations

Using critical point theory and monotonicity results, we consider the existence of solutions to certain types of difference equations depending on a parameter.     

First integrals of the equations ofnon-linear wave dynamics

In algorithm for binding the first integrals of fourth-order non-linear differential equations, encountered when describing non-linearwave processes, is process, is proposed. The use of the method is illustrated by a number of examples. The first integrals obtained are employed to construct a solution of the generalized fifth-order Korteweg-de Vries equation in travelling-wave variables, which is expressed in terms of hyperelliptic integrals.     

On the variational formulation of a transmission problem for the Biot equations

In this article, a variational formulation for the transmission problem of the fluid–bone interaction is formulated. The formulation is based on a modified Biot system of equations for the cancellous bone together with a boundary integral equation formulation of the pressure in the water. Existence and uniqueness for the weak solution of the interaction problem are established in appropriate Sobolev spaces.     

New criteria for exponential expansiveness of variational difference equations

The aim of this paper is to obtain general input-output conditions for uniform exponential expansiveness of variational difference equations in terms of the complete admissibility of pairs of sequence spaces. We introduce a large class Q(N) of Banach sequence spaces and we deduce the connections between the complete admissibility of the pair (B(Θ,V(N,X)),U(N,X)) with U,V∈Q(N) and the uniform exponential expansiveness of a system of variational difference equations. We apply our results at the study of the uniform exponential expansiveness of linear skew-product flows.     

Anti-periodic solutions of higher order nonlinear difference equations: a variational approach

A 2nth order nonlinear difference equation is studied. By using a variational approach, existence results for anti-periodic solutions are obtained under various assumptions upon the nonlinearities.     

Hamiltonian formulation of energy conservative variational equations by wavelet expansion

Hamiltonian formulation of various energy conservative evolution equations is given by means of wavelet expansion of solutions on the whole real axis R. The KdV equation, wave equations and Schrödinger equations are treated in a unified similar manner. A matrix representation of operators with respect to a nice wavelet base plays an important role in the formulation. Since the procedure is very concrete, our results can be used to efficiently compute numerical solutions of partial differential equations described in the text. In fact, we may also use symplectic schemes to solve derived Hamiltonian systems.     

Groups which do not admit ghosts

A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .

     

A variational formulation of anisotropic geometric evolution equations in higher dimensions

We present a novel variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow in , d ≥ 2. This new formulation leads to an unconditionally stable, fully discrete, parametric finite element approximation in the case d = 2 or 3. The resulting scheme has very good properties with respect to the distribution of mesh points and, if applicable, volume conservation. This is demonstrated by several numerical experiments for d = 3, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion.     

Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type,which do not possess Riemann invariants

We formulate several conjectures concerning the structure and properties of thenxn integrable nondiagonalizable hamiltonian systems of hydrodynamic type. Forn=3 our results are, in fact, complete: a 3×3 nondiagonalizable hamiltonian system is integrable if and only if it is weakly nonlinear (linearly degenerate).Institute for Mathematical Modelling, Academy of Sciences of Russia, 125047, Miusskaya, 4, Moscow, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 257–262, May, 1994.     

First integrals for time-dependent higher-order Riccati equations by nonholonomic transformation

We exploit the notion of nonholonomic transformations to deduce a time-dependent first integral for a (generalized) second-order nonautonomous Riccati differential equation. It is further shown that the method can also be used to compute the first integrals of a particular class of third-order time-dependent ordinary differential equations and is therefore quite robust.     

Relaxation results for a class of variational integrals

The relaxation problem for functionals of the form $\text{∝}{}_{\mathrm{\Omega }}\text{?(u, Du) dx}$ with $\text{?(s, z)}$ not necessarily continuous with respect to s is studied.     

Finite difference solution of Euler equations arising in variational image segmentation

This paper deals with finite-difference approximations of Euler equations arising in the variational formulation of image segmentation problems. We illustrate how they can be defined by the following steps: (a) definition of the minimization problem for the Mumford–Shah functional (MSf), (b) definition of a sequence of functionals Γ-convergent to the MSf, and (c) definition and numerical solution of the Euler equations associated to the kth functional of the sequence. We define finite difference approximations of the Euler equations, the related solution algorithms, and we present applications to segmentation problems by using synthetic images. We discuss application results, and we mainly analyze computed discontinuity contours and convergence histories of method executions. This revised version was published online in June 2006 with corrections to the Cover Date.     

First integrals and reduction of a class of nonlinear higher order ordinary differential equations

We propose a method for constructing first integrals of higher order ordinary differential equations. In particular third, fourth and fifth order equations of the form are considered. The relation of the proposed method to local and nonlocal symmetries are discussed.