  We study the octonionic selfduality equations for $p=3$-branes in the light
cone gauge and we construct explicitly, instanton solutions for spherical and
toroidal topologies in various flat spacetime dimensions $(D=5+1,7+1,8+1,9+1)$,
extending previous results for $p=2$ membranes. Assuming factorization of time
we reduce the self-duality equations to integrable systems and we determine
explicitly periodic, in Euclidean time, solutions in terms of the elliptic
functions. These solutions describe 4d associative and non-associative
calibrations in $D=7,8$ dimensions. It turns out that for spherical topology
the calibration is non compact while for the toroidal topology is compact. We
discuss possible applications of our results to the problem of 3-brane topology
change and its implications for a non-perturbative definition of the 3-brane
interactions.
