Symmetric hyperbolic systems include many physically relevant systems of partial differential equations (PDEs) such as Maxwell’s equations, the elastic wave equations and the acoustic equations [28]. In this paper we extend the Gaussian beam method to efficiently compute the high frequency solutions to such systems with polarized waves, in which the dispersion matrix of the hyperbolic system ha...

In this paper, we explore more applications of the hyperbolic function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations (PDES), to special discrete nonlinear equations. Some exact travelling wave solution of the discrete sine-Gordon equation are obtained in terms of hyperbolic fun...

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper, once named “On equations in relatively hyperbolic groups”. The first part is entitled “Existential questions in (relatively) hyperbolic groups” (to appear in Israel J. Math.). We study there the existential theory of torsion free hyperbolic and relatively hyperbolic group...

In this paper, we study the logarithmic stability for the hyperbolic equations by arbitrary boundary observation. Based on Carleman estimate, we first prove an estimate of the resolvent operator of such equation. Then we prove the logarithmic stability estimate for the hyperbolic equations without any assumption on an observation sub-boundary.

Recently, the study of initial-boundary value problems for hyperbolic equations with boundary integral conditions has received considerable attention. This kind of conditions has many important applications. For instance, they appear in the case where a direct measurement quantity is impossible; however, their mean values are known. In this paper, we deal with a class of quasilinear hyperbolic ...