New analytic chain soliton and self-similar solutions to the Kadomtsev-Petviashvili equation
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 has application to the modeling of weakly nonlinear fluid waves in the presence of gravity and depending on the sign of dispersion, can also describe a one parameter group of time dependent potentials of 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 Painlevé integrability criteria.