The present work extends the search of Jacobi elliptic function solutions for the multi-component modified Korteweg-de Vries equations. When the modulum m →1, those periodic solutions degenerate as the corresponding solitary wave and shock wave ones. Especially, exact solutions for the three-component system are presented in detail and graphically. 相似文献
A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.
AbstractFor the first order nonstationary hyperbolic equation taking the piecewise linear discontinuous Galerkin solver, we prove that under the uniform rectangular partition, such a discontinuous solver, after postprossesing, can have two and half approximative order which is half order higher than the optimal estimate by Lesaint and Raviart under the rectangular partition. 相似文献
Elementary symmetric polynomials can be thought of as derivative polynomials of . Their associated hyperbolicity cones give a natural sequence of relaxations for . We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves
hyperbolicity cones associated with elementary symmetric polynomials. As a consequence of this recursion, we give an alternative
characterization of these cones, and give an algebraic characterization for one particular dual cone associated with together with its self-concordant barrier functional. 相似文献
Sedimentation and erosion processes in sedimentary basins can be modeled by a parabolic equation with a limiter on the fluxes and a constraint on the time variation.This limiter happens to satisfy a stationary scalar hyperbolic inequality,within a constraint,for which the authors prove the existence and the uniqueness of the solution.Actually,this solution is shown to be the maximal element of a convenient convex set of functions.The existence proof is obtained thanks to the use of a numerical scheme. 相似文献
In this paper we investigate the support of the unique measure of maximal entropy of complex Hénon maps, J*. The main question is whether this set is the same as the analogue of the Julia set J.
July 4, 2005. The author is supported by an NSF grant 相似文献
This paper is devoted to strictly hyperbolic systems and equations with non‐smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of generalised functions. Extending earlier results on symmetric hyperbolic systems, we introduce generalised strict hyperbolicity, construct symmetrisers, prove an appropriate Gårding inequality and establish existence, uniqueness and regularity of generalised solutions. Under additional regularity assumptions on the coefficients, when a classical solution of the Cauchy problem (or of a transmission problem in the piecewise regular case) exists, the generalised solution is shown to be associated with the classical solution (or the piecewise classical solution satisfying the appropriate transmission conditions). 相似文献