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In quantum physics one finds the need to evaluate δ, when calculating the transition rates of certain particle interactions; see [1]. The problem of defining products of distributions is also closely connected with the problem of renormalization in quantum field theory. Due to the large use of distributions in the natural sciences and other mathematical fields, the problem of the product of dis...

A condition for existence of singular and delta shock waves for systems of conservation laws is given in the paper. The systems considered here have fluxes which are linear in one of the dependent variables. The condition obtained here is analogous to the one for the standard Hugoniot locus. Three different solution concept are used in the paper: associated solution in Colombeau sense, limits o...

It is well-known that distributional solutions to the Cauchy problem for ut +(b(t, x)u)x = 0 with b(t, x) = 2H(x− t), where H is the Heaviside function, are non-unique. However, it has a unique generalized solution in the sense of Colombeau. The relationship between its generalized solutions and distributional solutions is established. Moreover, the propagation of singularities is studied.

Bloch-periodicity is a generalization of the notion of periodic and antiperiodic functions with much practical relevance for engeneering science and especially condensed matter physics. In recent work we have considered this property in the setting of classical functional analysis, and also introduced the new notion of asymptotically Bloch-periodic functions. In this paper we formulate these pr...

This paper is devoted to the study of nonlinear geometric optics in Colombeau algebras of generalized functions in the case of Cauchy problems for semilinear hyperbolic systems in one space variable. Extending classical results, we establish a generalized variant of nonlinear geometric optics. As an application, a nonlinear superposition principle is obtained when distributional initial data ar...

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 logtype growth conditions on the s...

Stevan Pilipović, Mirjana Stojanović Abstract. We recall the definition of G–quasiasymptotics at infinity in a framework of Colombeau space G (cf. [8]) and give an application of that notion to a Cauchy problem for a strictly semilinear hyperbolic system. It turns out that quasiasymptotic behaviour at infinity of the solution inherits the quasiasymptotic behaviour at infinity of initial data un...