Evolution of lump solutions for the KP equation

The two (space)-dimensional generalisation of the Korteweg-de Vries (KdV) equation is the Kadomtsev-Petviashvili (KP) equation, This equation possesses two solitary wave type solutions. One is independent of the direction orthogonal to the direction of propagation and is the soliton solution of the KdV equation extended to two space dimensions. The other is a true two-dimensional solitary wave solution which decays to zero in all space directions. It is this second solitary wave solution which is considered in the present work. It is known that the KP equation admits an inverse scattering solution. However this solution only applies for initial conditions which decay at infinity faster than the reciprocal distance from the origin. To study the evolution of a lump-like initial condition, a group velocity argument is used to determine the direction of propagation of the linear dispersive radiation generated as the lump evolves. Using this information combined with conservation equations and a suitable trial function, approximate ODEs governing the evolution of the isolated pulse are derived. These pulse solutions have a similar form to the pulse solitary wave solution of the 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 (shedding mass) to a pulse of higher amplitude. The solutions of the approximate ODEs for the pulse evolution are compared with full numerical solutions of the KP equation and good agreement is found.