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(Applied Mathematics)

Faculty of Health, Engineering and Sciences

University of Southern Queensland

KP lumps: Structures, interactions, viscous decay

Exact solitary solutions to the Kadomtsev–Petviashvili (KP) equation with a positive dispersion are presented. The solutions represent completely localised two-dimensional formation in the form of obliquely propagating skew lumps. Specific features of such lumps are analysed in details. It is shown that skew multi-lump solutions can be also constructed within the framework of such an equation. As an example, the bi-lump solution is presented in the explicit form. The interactions of single lumps with each other and with multi-lumps, as well as the interactions of bi-lumps with each other are studied analytically and numerically. The results obtained are illustrated graphically and animated with the help of videos. The relevance of skew lumps to the real physical systems is discussed. The decay of lumps is also considered for few typical dissipation mechanisms, the Rayleigh dissipation, Reynolds dissipation, Chezy bottom friction, viscous dissipation in the laminar boundary layer, and radiative losses caused by large-scale dispersion. It is shown that the straight-line motion of lumps is unstable under the influence of dissipation. The lump trajectories are calculated for two most typical models of dissipation – the Rayleigh and Reynolds dissipations. A comparison of analytical results obtained within the framework of asymptotic theory with the direct numerical modelling within the KP equation is presented. Good agreement between the theoretical and numerical results is obtained.