Rationalsolutions of the KP equation through symmetry reductions:coherent structures and other solutions

Abstract

Some singular rational travelling wave solutions of theKadomtsev-Petviashvili (KP) equation are determined by makinguse of the theory of symmetry reductions. Solutions of KP of theform u(x-c1t,y-c2t), c1, c2 constants, satisfy theequation(-c1ūx̄-c2ūȳ + 6ūūx̄ + ūx̄x̄x̄)x̄ = ūȳȳ,with x̄ = x-c1t, ȳ = y-c2t andū(x̄,ȳ) = u(x-c1t,y-c2t). Some nonclassicalsymmetries of this equation are determined: then, by consideringthe corresponding ordinary differential equations, somesolutions for the KP equation are obtained. But in the case ofthe well known one-dimensional soliton, all the solutions weconstruct in this way are rational functions. Moreover, the KPequation reduces to the previous equation with c1 = c2 = 0 undera more general symmetry group. Thus, we can use the solutionsdescribing coherent structures to construct large families of(x,y)-rational solutions.