EVOLUTION OF LUMP SOLUTIONS FOR THE KP EQUATION

The two (space)-dimensional generalisation of the Korteweg-de Vries (K dV) equation is the Kadomtsev-Petviashvili (KP) equation, This equatio n possesses two solitary wave type solutions. One is independent of th e direction orthogonal to the direction of propagation and is the soli ton solution of the KdV equation extended to two space dimensions. The other is a true two-dimensional solitary wave solution which decays t o zero in all space directions. It is this second solitary wave soluti on which is considered in the present work. It is known that the KP eq uation admits an inverse scattering solution. However this solution on ly applies for initial conditions which decay at infinity faster than the reciprocal distance from the origin. To study the evolution of a l ump-like initial condition, a group velocity argument is used to deter mine the direction of propagation of the linear dispersive radiation g enerated as the lump evolves. Using this information combined with con servation equations and a suitable trial function, approximate ODEs go verning the evolution of the isolated pulse are derived. These pulse s olutions have a similar form to the pulse solitary wave solution of th e KP equation, but with varying parameters, It is found that the pulse solitary wave solutions of the KP equation are asymptotically stable, and that depending on the initial conditions, the pulse either decays to a pulse of lower amplitude (shedding mass) or narrows down (sheddi ng mass) to a pulse of higher amplitude. The solutions of the approxim ate ODEs for the pulse evolution are compared with full numerical solu tions of the KP equation and good agreement is found.