  Membranes in M-theory are expected to interact via splitting and joining
processes. We study these effects in the pp-wave matrix model, in which they
are associated with transitions between states in sectors built on vacua with
different numbers of membranes. Transition amplitudes between such states
receive contributions from BPS instanton configurations interpolating between
the different vacua. Various properties of the moduli space of BPS instantons
are known, but there are very few known examples of explicit solutions. We
present a new approach to the construction of instanton solutions interpolating
between states containing arbitrary numbers of membranes, based on a continuum
approximation valid for matrices of large size. The proposed scheme uses
functions on a two-dimensional space to approximate matrices and it relies on
the same ideas behind the matrix regularisation of membrane degrees of freedom
in M-theory. We show that the BPS instanton equations have a continuum
counterpart which can be mapped to the three-dimensional Laplace equation
through a sequence of changes of variables. A description of configurations
corresponding to membrane splitting/joining processes can be given in terms of
solutions to the Laplace equation in a three-dimensional analog of a Riemann
surface, consisting of multiple copies of R^3 connected via a generalisation of
branch cuts. We discuss various general features of our proposal and we also
present explicit analytic solutions.
