Linearization and exact solutions of a class of Boltzmann equations

A Boltzmann equation with scattering cross section that depends on the reciprocal and first power of the modulus of the relative velocity is considered. It is solved by application of a Fourier transformation with respect to the velocity and then separation of the variables. The obtained infinite hierarchy of nonlinear ordinary differential equations is solved by means of the method of Poincaré normal forms, generalized to the infinite-dimensional case. Convergence of the series that represents the solution is proved.