On the local and global well-posedness theory for the KP-I equation

In this paper we obtain new local and global well-posedness results for the KP-I equation.

Résumé

Dans cet article nous obtenons de nouveaux résultats sur le caractère bien posé local et global de l'équation KP-I.     

Gain of regularity for the KP-I equation

In this paper we study the smoothness properties of solutions to the KP-I equation. We show that the equation's dispersive nature leads to a gain in regularity for the solution. In particular, if the initial data ? possesses certain regularity and sufficient decay as x→∞, then the solution u(t) will be smoother than ? for 0<t?T where T is the existence time of the solution.     

Approximated Vector Potential of Intense Laser Pulses in a Pair Plasma

A （3＋1 ）-dimensional Kadomtse-Petviashvili （KP） equation for nonlinearly interacting intense laser pulses with an electron-positron （e-p） plasma is derived. Taking into account the combined action of the relativistic particle mass increase and the relativistic light ponderomotive force, using the perturbation method, and allowing different types solution, we discuss the analytical solution of （3＋1）-dimensional KP-I equation, and give the approximate solutions of vector potential of the intense laser pulse in e-p plasma. Our results may be significantly useful in understanding the nonlinear wave propagation and interaction of intense laser beams in an e-p plasma.