高阶拟线性双曲型方程的精确边界能控性

By means of the existence and uniqueness of semi-global C^1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues ,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.     

Hyperbolic equations with third-order symmetries

We present a complete list of nonlinear one-field hyperbolic equations with integrable third-order x and y symmetries. The list includes both equations of the sine-Gordon type and equations linearizable by differential substitutions.     

Transport equations of higher order discontinuities for quasilinear hyperbolic systems

The transport equations satisfying ordinary linear differential equations of first order which govern the behaviour of higher order discontinuities for quasilinear hyperbolic systems along the rays associated with a singular surface are derived. It is shown that the transport equations depend on the Gaussian curvature of wave front.     

Reduction and quotient equations for differential equations with symmetries

The method of reduction previously known in the theory of Hamiltonian systems with symmetries is developed in order to obtain exact group-invariant solutions of systems of partial differential equations. This method leads to representations of quotient equations which are very convenient for the systematic analysis of invariant solutions of boundary value problems. In the case of partially invariant solutions, necessary and sufficient conditions of their invariance with respect to subalgebras of symmetry algebras are given. The concept of partial symmetries of differential equations is considered.     

On energy estimates for second order hyperbolic equations with Levi conditions for higher order regularity

We consider energy estimates for second order homogeneous hyperbolic equations with time dependent coefficients. The property of energy conservation, which holds in the case of constant coefficients, does not hold in general for variable coefficients; in fact, the energy can be unbounded as t → ∞ in this case. The conditions to the coefficients for the generalized energy conservation (GEC), which is an equivalence of the energy uniformly with respect to time, has been studied precisely for wave type equations, that is, only the propagation speed is variable. However, it is not true that the same conditions to the coefficients conclude (GEC) for general homogeneous hyperbolic equations. The main purpose of this paper is to give additional conditions to the coefficients which provide (GEC); they will be called as C ^k -type Levi conditions due to the essentially same meaning of usual Levi condition for the well-posedness of weakly hyperbolic equations.     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Discretization of second-order ordinary differential equations with symmetries

A number of publications (indicated in the Introduction) are overviewed that address the group properties, first integrals, and integrability of difference equations and meshes approximating second-order ordinary differential equations with symmetries. A new example of such equations is discussed in the overview. Additionally, it is shown that the parametric families of invariant difference schemes include exact schemes, i.e., schemes whose general solution coincides with the corresponding solution set of the differential equations at mesh nodes, which can be of arbitrary density. Thereby, it is shown that there is a kind of mathematical dualism for the problems under study: for a given physical process, there are two mathematical models: continuous and discrete. The former is described by continuous curves, while the latter, by points on these curves.     

Non-local symmetries of first-order equations

It is shown how one can transform scalar first-order ordinarydifferential equations which admit non-local symmetries of theexponential type to integrable equations admitting canonicalexponential non-local symmetries. As examples we invoke theAbel equation of the second kind, the Riccati equation and naturalgeneralizations of these. Moreover, our method describes howa double reduction of order for a second-order ordinary differentialequation which admits a two-dimensional Lie algebra of generatorsof point symmetries can be affected if the second-order equationis first reduced in order once by a symmetry which does notspan an ideal of the two-dimensional Lie algebra.     

Quasi-linear weakly hyperbolic equations with Gevrey-Levi conditions

This paper is devoted to the study of the analytic regularity for real solutions of a non-linear weakly hyperbolic equation of the form: $$\sum\limits_{m - (1 - e)r< |a| \leqslant m} {a_a (y,u^{(\beta )} } )\partial _y^a u = g(y,u^{(\beta )} ){\text{, |}}\beta | \leqslant m - (1 - e)r,$$ wherea _α andg are analytic functions of their arguments,r≥2 denotes the largest multiplicity of the characteristic roots of (*) and ?∈(0,(r?1)/r). Assuming that 026 the characteristic roots are of constant multiplicity and that the generalized Levi’s conditions related to the index ? are satisfied, we prove that, ifu is a solution of (*) 026 and belongs to a Gevrey class of index σ<1/?, then the analyticity of Cauchy data propagates according to the geometry of the influence domains of the equation.     

First-order hyperbolic pseudodifferential equations with generalized symbols

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators [J. Math. Anal. Appl. 160 (1991) 93-106, J. Math. Anal. Appl. 142 (1989) 452-467].