  We formulate the notion of the classical
Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) reaction diffusion equation
associated with a homogeneous conservative fragmentation process and study its
traveling waves. Specifically, we establish existence, uniqueness and
asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl.
Math. 28 (1975) 323-331] and [Comm. Pure Appl. Math. 29 (1976) 553-554], Neveu
[In Seminar on Stochastic Processes (1988) 223-242 Birkh\"{a}user] and Chauvin
[Ann. Probab. 19 (1991) 1195-1205], our analysis exposes the relation between
traveling waves and certain additive and multiplicative martingales via laws of
large numbers which have been previously studied in the context of
Crump-Mode-Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57
(1981) 365-395] and in the context of fragmentation processes by Bertoin and
Martinez [Adv. in Appl. Probab. 37 (2005) 553-570] and Harris, Knobloch and
Kyprianou [Ann. Inst. H. Poincar\'{e} Probab. Statist. 46 (2010) 119-134]. The
conclusions and methodology presented here appeal to a number of concepts
coming from the theory of branching random walks and branching Brownian motion
(cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503-517] and Biggins
and Kyprianou [Electr. J. Probab. 10 (2005) 609-631]) showing their
mathematical robustness even within the context of fragmentation theory.

Probability;Algebraic Geometry