New analytic chain soliton and self-similar solutions to the Kadomtsev-Petviashvili equation
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The KP (Kadomtsev-Petviashvili) equation is probably the most studied (2+1)-dimensional integrable wave equation of the known (2+1)-dimensional integrable models. The KP equation is often used with positive dispersion (denoted as KP-2), to describe weakly nonlinear shallow water waves with one dominant wave vector, and the KP equation with negative dispersion (denoted as KP-1) gives a one parameter group of time dependent potentials corresponding to the Schrodinger equation.
In this work, the exact nonlinear instability of the KP-1 equation is described where an unstable soliton decays into a slow soliton and fast ‘chain’ soliton. Certain aspects of the behaviour of the solutions are explored, such as, branching of the periodic chain solitons. And finally, the self-similar solution to the KP equation is constructed and the resulting ODE is related to the cylindrical KDV equation as well as the Painleve integrability criteria.