Abstract :
The propagation of travelling waves is a relevant physical phenomenon. As usual the understanding
of a real propagating wave depends upon a correct formulation of a idealized model.
Discontinuous functions, Dirac-δ measures and their distributional derivatives are, respectively, idealizations
of sharp jumps, localized high peaks and single sharp localised oscillations. In the present
paper we study the propagation of distributional travelling waves for Burgers inviscid equation. This
will be afforded by our theory of distributional products, and is based on a rigorous and consistent
concept of solution we have introduced in [C.O.R. Sarrico, Distributional products and global solutions
for nonconservative inviscid Burgers equation, J. Math. Anal. Appl. 281 (2003) 641–656].
Our approach exhibit Dirac-δ travelling solitons (they are just the “infinitesimal narrow solitons”
of Maslov, Omel’yanov and Tsupin [V.P. Maslov, O.A. Omel’yanov, Asymptotic soliton-form solutions
of equations with small dispersion, Russian Math. Surveys 36 (1981) 73–149; V.P. Maslov,
V.A. Tsupin, Necessary conditions for the existence of infinitely narrow solitons in gas dynamics,
Soviet Phys. Dokl. 24 (1979) 354–356]) and also solutions which are not measures such as for instance
u(x, t) = b + δ
(x − bt), a wave of constant speed b. Moreover, for signals with two jump
discontinuities we have, in our setting, the propagation of more solitons and more values for the
signal speed are allowed than those afforded within classical framework.
© 2005 Elsevier Inc. All rights reserved.
Keywords :
Products of distributions , Propagation of distributional signals , Delta-solitons , Delta-waves , Nonconservative Burgers equation
