Abstract:
Via the elementary Darboux transformation (DT) of the modified Kadomtsev--Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.

Abstract:
We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the $n$-th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy we get the line soliton, the lump solution and the Johnson-Thompson lump, and the corresponding determinant formulae for the non-linear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases.

Abstract:
In the paper we investigate integrability characteristics for the dispersionless Kadomtsev-Petviashvili hierarchy. These characteristics include symmetries, Hamiltonian structures and conserved quantities. We give a Lax triad to construct a master symmetry and a hierarchy of non-isospectral dispersionless Kadomtsev-Petviashvili flows. These non-isospectral flows, together with the known isospectral dispersionless Kadomtsev-Petviashvili flows, form a Lie algebra, which is used to derive two sets of symmetries for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy. By means of the master symmetry, symmetries, Noether operator and conserved covariants, Hamiltonian structures are constructed for both isospectral and non-isospectral dispersionless Kadomtsev-Petviashvili hierarchies. Finally, two sets of conserved quantities and their Lie algebra are derived for the isospectral dispersionless Kadomtsev-Petviashvili hierarchy.

Abstract:
We construct the formal solution of the Cauchy problem for the dispersionless Kadomtsev - Petviashvili equation as application of the Inverse Scattering Transform for the vector field corresponding to a Newtonian particle in a time-dependent potential. This is in full analogy with the Cauchy problem for the Kadomtsev - Petviashvili equation, associated with the Inverse Scattering Transform of the time dependent Schroedinger operator for a quantum particle in a time-dependent potential.

Abstract:
An asymptotic behaviour of solution of Kadomtsev-Petviashvili-2 equation is obtained as $t\to\infty$ uniformly with respect to spatial variables.

Abstract:
The coupled Kadomtsev-Petviashvili equation is considered. It is shown that a Darboux transformation can be constructed by means of an elementary approach.

Abstract:
An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev--Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painleve' I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture.

Abstract:
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--Petviashvili equation. Since many classic high order splitting methods do not perform well, we propose a stable extrapolation method in order to construct an efficient numerical scheme of order four. In addition, the conservation properties and the possibility of order reduction for certain initial values for the numerical schemes under consideration is investigated.

Abstract:
This work describes a classification of the $N$-soliton solutions of the Kadomtsev-Petviashvili II equation in terms of chord diagrams of N chords joining pairs of 2N points. The different classes of N-solitons are enumerated by the distribution of crossings of the chords. The generating function of the chord diagrams is expressed as a continued fraction, special cases of which are moment generating functions for certain kinds of $q$-orthogonal polynomials.

Abstract:
Many equations possess soliton resonances phenomenon, this paper studies the soliton resonances of the nonisospectral modified Kadomtsev-Petviashvili (mKP) equation by asymptotic analysis.