A study of the nonlinear breakage equation: analytical and asymptotic solutions

New solutions of the nonlinear (collisional) breakage equation are given using analytical and asymptotic methods. The dynamic nonlinear breakage equation is transformed to a linear one for some simple forms of the collision kernel; methods for treating the linear equation are employed to obtain solutions for the nonlinear case. Furthermore, it is shown that under particular conditions the particle size distribution can take asymptotically a self-similar form, i.e. the shape of the (appropriately normalized) distribution is independent of time. The self-similar distribution is obtained from the solution of a double nonlinear integral equation. The latter is solved in closed form and numerically (after transformation to a boundary value problem) for simple forms of the collision and breakage kernels; results for the self-similar distribution are presented and discussed.