Here u = u(x, y, t) is a scalar function, x and y are respectively the longitudinal and transverse spatial coordinates, subscripts x, y, t denote partial derivatives, and σ2 = ±1. The case σ = 1 is known as the KPII equation, and models, for instance, water waves with small surface tension. The case σ = i is known as the KPI equation, and may be used to model waves in thin films with high surfa...

We consider the (KdV)/(KP-I) asymptotic regime for the Nonlinear Schrödinger Equation with a general nonlinearity. In a previous work, we have proved the convergence to the Korteweg-de Vries equation (in dimension 1) and to the Kadomtsev-Petviashvili equation (in higher dimensions) by a compactness argument. We propose a weakly transverse Boussinesq type system formally equivalent to the (KdV)/...

An asymptotic behaviour of solution of Kadomtsev-Petviashvili-2 equation is obtained as t → ∞ uniformly with respect to spatial variables.

We consider a splitting approach for the Kadomtsev–Petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented. The error made by this numerical scheme is compared to exponential integrators which have been shown in Klein and Roidot (SIAM J. Sci. Comput., 2011) to perform best for stiff solutions of the Kadomtsev–Pet...

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili [Kadomtsev BB, Petviashvili VI (1970) Sov Phys Dokl 15:539-541] proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solution...

The spectrum of a Schrödinger operator with a perfect honeycomb lattice potential has special points, called Dirac points where the lowest two branches of the spectrum touch. Deformations can result in the merging and disappearance of the Dirac points and the originally intersecting dispersion relation branches separate. Corresponding to these deformations, nonlinear envelope equations are deri...

In this work, we are mainly concerned with the existence of stationary solutions for the generalized Kadomtsev-Petviashvili equation in bounded domain in R

We prove that the flow map of the Kadomtsev-Petviashvili-I (KP-I) equation is not uniformly continuous on bounded sets of the natural energy space.

Bäcklund transformations, which are relations among solutions of partial differential equations–usually nonlinear–have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables, but they are rare. Wahlquist and Estabrook [2] discovered a systematic method for searchi...