Symmetries of compatibility conditions for systems of differential equations

When symmetries of differential equations are applied, various types of associated systems of equations appear. Compatibility conditions of the associated systems expressed in the form of differential equations inherit Lie symmetries of the initial equations. Invariant solutions to compatibility systems are known as orbits of partially invariant and generic solutions involved in the Lie group foliation of differential equations and so on. In some cases Bäcklund transformations and differential substitutions connecting quotient equations for compatibility conditions and initial systems naturally arise. Besides, Ovsiannikov's orbit method for finding partially invariant solutions is essentially based on such symmetries.     

Partial differential equations with differential constraints

A geometric setting for constrained exterior differential systems on fibered manifolds with n-dimensional bases is proposed. Constraints given as submanifolds of jet bundles (locally defined by systems of first-order partial differential equations) are shown to carry a natural geometric structure, called the canonical distribution. Systems of second-order partial differential equations subjected to differential constraints are modeled as exterior differential systems defined on constraint submanifolds. As an important particular case, Lagrangian systems subjected to first-order differential constraints are considered. Different kinds of constraints are introduced and investigated (Lagrangian constraints, constraints adapted to the fibered structure, constraints arising from a (co)distribution, semi-holonomic constraints, holonomic constraints).     

Isogroups of differential ideals of vector-valued differential forms: Application to partial differential equations

An older geometric technique for the study of invariance groups of partial differential equations, originally proposed by one of the authors and F. B. Estabrook, is generalized and extended to problems involving exterior equations for vector-valued or Lie algebra-valued exterior differential forms. Use of the method is demonstrated in the study of the symmetry groups of the two-dimensional Dirac equation and the full Yang-Mills free-field equations in Minkowski spacetime.     

Persistence properties for a family of nonlinear partial differential equations

We examine the persistence of decay properties for a family of dispersive nonlinear partial differential equations. We show that certain decay properties of the initial data persist for as long as the solution exists. On the other hand, for a subset of the family certain decay rates are possible only for the trivial solution. For example, the only solution that remains with compact support for any further time is the trivial solution.     

Wong-Zakai approximations for stochastic differential equations

The aim of this paper is to give a wide introduction to approximation concepts in the theory of stochastic differential equations. The paper is principally concerned with Zong-Zakai approximations. Our aim is to fill a gap in the literature caused by the complete lack of monographs on such approximation methods for stochastic differential equations; this will be the objective of the author's forthcoming book. First, we briefly review the currently-known approximation results for finite- and infinite-dimensional equations. Then the author's results are preceded by the introduction of two new forms of correction terms in infinite dimensions appearing in the Wong-Zakai approximations. Finally, these results are divided into four parts: for stochastic delay equations, for semilinear and nonlinear stochastic equations in abstract spaces, and for the Navier-Stokes equations. We emphasize in this paper results rather than proofs. Some applications are indicated.The author's research was partially supported by KBN grant No. 2 P301 052 03.     

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.     

Deformations of nonassociative algebras and integrable differential equations

A new class of nonassociative algebras related to integrable PDE's and ODE's is introduced. These algebras can be regarded as a noncommutative generalization of Jordan algebras. Their deformations are investigated. Relationships between such algebras and graded Lie algebras are established.     

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.     

Semi-integer derivatives of the Airy functions and related properties of the Korteweg-de Vries-type equations

Fractional integrals and derivatives of Airy functions (Riesz potentials) are considered. For half integrals D ^−1/2 Ai(x) and D ^−1/2 Gi(x) explicit representations are found in terms of the products of Airy functions. Here Ai(x) and Gi(x) are the Airy function of the first kind and the Scorer function, respectively. Based on that representations are obtained for all semi-integer derivatives of Ai(x) and Gi(x). Applications to Korteweg–de Vries type equations are provided.      

Symmetries of distributions and quadrature of ordinary differential equations

We present a geometric exposition of S. Lie's and E. Cartan's theory of explicit integration of finite-type (in particular, ordinary) differential equations. Numerous examples of how this theory works are given. In one of these, we propose a method of hunting for particular solutions of partial differential equations via symmetry preserving overdetermination.     

Rigorous derivation of incompressible type Euler equations from non-isentropic Euler-Maxwell equations

In this paper, we investigate the convergence of the time-dependent and non-isentropic Euler-Maxwell equations to incompressible Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and energy method.     

On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations

The paper is devoted to studying the image of probability measures on a Hilbert space under finite-dimensional analytic maps. We establish sufficient conditions under which the image of a measure has a density with respect to the Lebesgue measure and continuously depends on the map. The results obtained are applied to the 2D Navier-Stokes equations perturbed by various random forces of low dimension.     

Sobolev type embedding and quasilinear elliptic equations with radial potentials

We study the existence of nontrivial radial solutions for quasilinear elliptic equations with unbounded or decaying radial potentials. The existence results are based upon several new embedding theorems we establish in the paper for radially symmetric functions.     

Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|^p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|⁻² where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.     

Nonlinear scattering for some dispersive equations generalizing Benjamin-Bona-Mahony equation

The time decay of solutions to nonlinear dispersive equations of the typeMu _t+F(u)_x=0 is established using the optimal estimates for the linearized equation and standard techniques from scattering theory.     

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.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.