Kadomtsev-Petviashvili equation

Abstract

The KP equation is a universal integrable system in two spatial dimensions in the same way that the KdV equation can be regarded as a universal integrable system in one spatial dimension, since many other integrable systems can be obtained as reductions. As such, the KP equation has been extensively studied in the mathematical community in the last forty years. The KP equation is also one of the most universal models in nonlinear wave theory, which arises as a reduction of system with quadratic nonlinearity which admit weakly dispersive waves, in a paraxial wave approximation. The equation naturally emerges as a distinguished limit in the asymptotic description of such systems in which only the leading order terms are retained and an asymptotic balance between weak dispersion, quadratic nonlinearity and diffraction is assumed. The different role played by the two spatial variables accounts for the asymmetric way in which they appear in the equation. Despite their apparent similarity, the two versions of the KP equation differ significantly with respect to their underlying mathematical structure and the behavior of their solutions. Figure 1 shows a two-dimensional localized solution of the KPI equation, known as the lump solution, while Figure 2 shows a contour plot of a resonant two-soliton solution of the KPII equation. Solutions of these KP equations and their properties are discussed in more detail in the following sections.