Some Approaches to Solving the KP Equation: Different Representations and Various Types of Solutions

We present different methods to construct solutions to the Kadomtsev–Petviashvili (KP) equation. In the first method, from the solutions to the NLS equation, we construct solutions to the KP equation in terms of Fredholm determinants. We deduce solutions written as quotients of Wronskians of order 2N. When one of these parameters tends to zero, we obtain N-order rational solutions expressed as a quotient of two polynomials of degree 2N(N+1) in x, y and t, depending on 2N−2 real parameters. We obtain, in this case, regular solutions to the KP equation. Using new results from the NLS equation, with solutions constructed in terms of quotients of determinants of order N depending on 2N−2 real parameters, we are able to highlight new forms of configurations, such as triangles and concentric rings. Another approach using the Darboux transformation is given to get multi-parametric solutions to the KP equation. In this approach, it is possible to construct an infinite hierarchy of solutions depending on the degree of summation and the degree of derivation. The third method involves choosing special polynomials and using the bilinear Hirota method to get other types of solutions to the KP equation. We also obtain an infinite hierarchy of solutions depending on the order of the determinants. The last method allows the construction of regular solutions to the KP equation. In this approach, we obtain another alternative to obtain regular solutions, as in the case of the first method, and we also observe the formation of configurations such as triangles or concentric rings. We study the configurations of these hierarchies of solutions to the KP equation as a function of their different parameters.