### Nonlinear Wave Propagation in Media with Dispersion and Dissipation''

Abstract: George Boussinesq was the first to explain scientifically the effect of existence of solitary waves or solitons discovered more than thirty years earlier by the famous Scottish physicist and engineer Scott Russell. In 1871-77 in a series of papers Boussinesq deduced a semilinear evolution equation governing the propagation of long waves of small amplitude over the surface of shallow water. In contrast to the famous Kortweg-de Vries equation it is of the second order in time and can be written as

u_tt = - \alpha u_xxxx + u_xx + \beta(u^2)_xx ,

where \beta is a real number, \alpha<0 corresponds to the original equation derived by the author (called later the "bad" Boussinesq equation because of its linear stability) and \alpha>0 corresponds to the so called "good" Boussinesq equation related to the small nonlinear oscillations of elastic rods. The latter one has linear stability and is quite attractive to mathematicians for this reason. We shall deal with the Cauchy problem for the "good" Boussinesq equation with dissipation, that is one more term is appended to the left-hand side, - 2bu_txx, where b>0. This term accounts for internal damping. The issues of global-in-time existence, uniqueness and construction of solutions will be studied.

Another interesting model we would like to consider is related to the well known Kadomtsev-Petviashvili equation, but in contrast to the latter one it exists in both one-dimensional and two-dimensional versions. It is called the Ostrovsky equation, and it takes into account rotation (Coriolis effect). We shall consider the Cauchy problem for the one-dimensional Ostrovsky equation governing long waves of small amplitude propagating to the right, construct its global-in-time solutions and establish their long-time behavior. In this problem dissipation is not taken into account, and dispersive effects determine the time decay.