Nonlocal trends in the geometry of differential equations: Symmetries,conservation laws,and Bäcklund transformations

The theory of coverings over differential equations is exposed which is an adequate language for describing various nonlocal phenomena: nonlocal symmetries and conservation laws, Bäcklund transformations, prolongation structures, etc. A notion of a nonlocal cobweb is introduced which seems quite useful for dealing with nonlocal objects.     

Decay of conservation laws and their generating functions

Quantities which are conserved in nondissipative media decay in the presence of dissipation. The velocity of such a decay (or the balance law) may be found explicitly using the generating function of the conservation law. The general approach is illustrated with a system of MHD equations for incompressible magnetofluids.     

Local symmetries and conservation laws

Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of higher KdV equations are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that symmetry and conservation law are, in some sense, the dual conceptions which coincides in the self-dual case, namely, for Euler-Lagrange equations. Training examples are also given.Translated from the Russian by B. A. Kuperschmidt.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

Symmetries and conservation laws of partial differential equations: Basic notions and results

The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.     

Arbitrariness of the general solution and symmetries

The computation of a number of arbitrary functions in the general solution is briefly reviewed. The results are used to study normal systems and their symmetry reduction. We discuss the treatment of gauge systems, especially the analysis of gauge fixing conditions. As examples, the Yang-Mills equations with the Lorentz gauge and Einstein's vacuum field equations with harmonic coordinates are considered.     

Symmetric boundary values for the Dirichlet problem for harmonic maps from the disc into the 2-sphere

Let Aut(D) denote the group of biholomorphic diffeormorphisms from the unit disc D onto itself and O(3) the group of orthogonal transformations of the unit sphere S ². The existence of multiple solutions to the Dirichlet problem for harmonic maps from D into S ² is related to the symmetries (if any) of the boundary value γ : ∂D → S ², by invariance of the Dirichlet energy under the action of Aut(D) × O(3). In this paper, we classify the stabilizers in Aut(D) × O(3) of boundary values in H ^1/2(S ¹, S ²) and . We give two applications to the Dirichlet problem for harmonic maps. This work was partially supported by the CMLA, Ecole Normale Supérieure de Cachan, Cachan, France.     

On thec-spectral sequence for systems of evolution equations

The groups E ₁ ^2,n–1 ( ) of Vinogradov'sC-spectral sequence for determined systems of evolution equations are considered. Presentation of these groups useful in practical computations is obtained. The group E ₁ ^2,1 ( ) is calculated for a system of Schrödinger type equations.     

Graded differential equations and their deformations: A computational theory for recursion operators

An algebraic model for nonlinear partial differential equations (PDE) in the category ofn-graded modules is constructed. Based on the notion of the graded Frölicher-Nijenhuis bracket, cohomological invariants H ^* (A) are related to each object (A, ) of the theory. Within this framework, H ⁰ (A) generalizes the Lie algebra of symmetries for PDE's, while H ¹ (A) are identified with equivalence classes of infinitesimal deformations. It is shown that elements of a certain part of H ¹ (A) can be interpreted as recursion operators for the object (A, ), i.e. operators giving rise to infinite series of symmetries. Explicit formulas for computing recursion operators are deduced. The general theory is illustrated by a particular example of a graded differential equation, i.e. the Super KdV equation.Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia.     

Local well-posedness of a new integrable equation

In this article we focus on the local-in-time well-posedness of the Cauchy problem for a new integrable equation. We proved the local-in-time existence and uniqueness of the entropy solutions by using the method of the vanishing viscosity and L¹$L^1$-contraction property.     

A method for computing symmetries and conservation laws of integra-differential equations

A method for computing symmetries and conservation laws of integro-differential equations is proposed. It resides in reducing an integro-differential equation to a system of boundary differential equations and in computing symmetries and conservation laws of this system. A geometry of boundary differential equations is constructed like the differential case. Results of the computation for the Smoluchowski's coagulation equation are given.     

Local nonintegrability of long-short wave interaction equations

The algebra of higher symmetries and the space of conservation laws for Zakharov's nonlinear equations of the interaction between long and short waves are completely described. The scheme of computations due to Vinogradov is used. As a result, the local nonintegrability of these equations is proved.     

Symmetries,invariant solutions and conservation laws of the nonlinear acoustics equation

The symmetry algebra of the Khoklov-Zabolotskaya equation is found,n- and (n-1)-dimensional subalgebrasL are classified (n is an independent variable number) andL-invariant solutions described. Conservation laws and conserved flows are also found.     

Spencer cohomologies and symmetry groups

We propose the construction of a spectral sequence converging to Spencer cohomologies. By using symmetry groups of differential equations systems, we manage to unify computations by reduction to the invariant systems over a homogeneous space. The conditions of coincidence of Spencer cohomologies with the cohomologies of an invariant Spencer complex we obtain from the arithmetic of a -characteristic manifold with respect to fundamental weights of the homogeneous space.     

A density function and the structure of singularities of the mean curvature flow

We study singularity formation in the mean curvature flow of smooth, compact, embedded hypersurfaces of non-negative mean curvature in ⁿ⁺¹, primarily in the boundaryless setting. We concentrate on the so-called Type I case, studied by Huisken in [Hu 90], and extend and refine his results. In particular, we show that a certain restriction on the singular points covered by his analysis may be removed, and also establish results relating to the uniqueness of limit rescalings about singular points, and to the existence of slow-forming singularities of the flow.The main new ingredient introduced, to address these issues, is a certain density function, analogous to the usual density function in the study of harmonic maps in the stationary setting. The definition of this function is based on Huisken's important monotonicity formula for mean curvature flow.     

Estimates for eigenvalues on Riemannian manifolds

In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space Rn, the conjecture of Pólya is well known: the kth eigenvalue λk of the Dirichlet eigenvalue problem of Laplacian satisfies

     

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235-1270], especially, the blow-up of weak solutions in the case of non-negative energy.     

Bounds on eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space R ⁿ . If λ _k+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥ 41 and and, for any n and with , where j _p,k denotes the k-th positive zero of the standard Bessel function J _p (x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k.Q.-M. Cheng was partially Supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of ScienceH. Yang was partially Supported by Chinese NSF, SF of CAS and NSF of USA     

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic-elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L¹-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic-parabolic and the hyperbolic-hyperbolic relaxation limits for that system.