Recent results on the generalized Kadomtsev-Petviashvili equations
This paper presents a concise review of rigorous results obtained for the generalized Kadomtsev-Petviashvili (KP) equations in two dimensions, \[ u_t+ f(u) u_x+ u_{xxx}+ \varepsilon v_y= 0,\quad v_x= u_y,\tag{1} \] (where \(\varepsilon= \pm 1\)) and for similar equations in three dimensions. The power nonlinearities, \(f(u)= u^p\), are considered. It is known that the two-dimensional KP equations with \(p= 1\) are integrable by means of the inverse scattering transform. In this context, the KP equations with \(\varepsilon= -1\) and \(+1\) are called, respectively, the KP-I and KP-II equations. Nevertheless, even in this special case traditional methods of the PDE theory prove to be more powerful than the inverse scattering for proving the basic theorems about existence and uniqueness of solutions to the Cauchy problem for the KP equations. In particular, the global well-posedness of the Cauchy problem is proven for the integrable KP-II case, while for the KP-I case it is proven only for small initial data. Another important set of results is known for blowup (collapse) of the solutions, i.e., singularity formation in a finite time. While for the KP-II type equations this problem remains open, for the KP-I type using the so-called virial theorem (in two dimensions) has demonstrated that the blowup does take place if \(p\geq 4/3\), and does not happen in the opposite case. The final group of results pertains to existence and stability of solitons, i.e., solutions localized in all the spatial directions (note that the multidimensional KP solitons are localized not exponentially but only algebraically). It proves that the soliton solutions to the generalized KP equations can be completely classified. In particular, a soliton exists as a solution to the three-dimensional KP-I with \(p= 1\), but it is unstable. Generally, instability of the solitons is closely linked to the possibility of the blowup in the system.
global well-posedness of the Cauchy problem
scientific article; zbMATH DE number 790814
scientific article