On a coupled system of KP-type
Jacob B. Aguilar
TL;DR
The paper introduces the Non-KP model, a coupled KP-type system derived via a variational approach from the water-wave problem to avoid the KP zero-mass constraint. It establishes the Hamiltonian structure, demonstrates mass conservation, and diagonalizes the linear part to reveal two dispersive modes with nonlinear couplings. By developing function spaces tailored to the system and proving linear estimates in time-localized Bourgain spaces, it lays a rigorous foundation for well-posedness and further dispersive analysis of the Non-KP model. The work integrates variational derivation, Hamiltonian formalism, diagonalization, and $X^{s,b}$-type estimates to advance the analytical understanding of coupled KP-type dynamics in a dispersive setting.
Abstract
A defining characteristic of the Kadomstev-Petviashvili (KP) model equation is that the well-posedness results are subject to the restriction that at all transverse positions, the mass $\int u \,dx = \text{constant independent of $y$}.$ In 2007, for a rather general class of equations of KP type, it was shown that the zero-mass (in $x$) constraint is satisfied at any non-zero time even if it is not satisfied at initial time zero. To remedy this ``odd'' behavior, a model modification is introduced which does not impose non-physical restrictions upon the initial data. In this article, we introduce a new modified KP system, named the Non-KP model equation. After providing a variational derivation of the Non-KP model, we analyze its Hamiltonian evolutionary structure. Furthermore, we prove linear estimates in the Bourgain spaces $X^{s,b}$ corresponding to the integral equation arising from the Duhamel formulation of system.