Kadomtsev-Petviashvili equation

The Kadomtsev-Petviashvili equations (KP) are given by

$(u_{t}+u_{xxx}^{}+uu_{x})_{x}+\lambda u_{yy}=0$

where $u=u(x,y,t)$ is a real-valued function of two space variables and one time variable, and $\lambda$ is a constant scalar. When $\lambda =0$ the evolution is trivial in the y variable and the equation collapses to the KdV equation. When $\lambda <0$ the equation is known as the KP-I equation, and when $\lambda >0$ the equation is the KP-II equation. These equations are a model for shallow long waves in the x direction, with some mild dispersion in the y direction (KdPv1970, PvYn1989). Generally speaking, KP-I is a good model when surface tension is strong, and KP-II is a good model when surface tension is weak.

KP (or the 2D Boussinesq equations) can also arise as a model for the (somewhat unphysical) 3D water wave equation in the shallow case, similar to how KdV arises from the 2D water wave equation. Under further limiting assumptions (slowly varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least). KP-I also arises as a model for sound waves in ferromagnetic media (TrFl1985).

The equations are completely integrable, and thus have an infinite number of conserved quantities. However for the LWP and GWP theory the most important conserved quantities are the mass

$\int u^{2}dxdy$

and the Hamiltonian

$\int u_{x}^{2}-u^{3}/3-\lambda (\nabla _{x}^{-1}u_{y})^{2}dxdy.$

The next conserved quantity contains terms which resemble the L^2 norm of u_xx or of $\nabla _{x}^{-2}u_{yy}$. (Note that one y derivative has the same scaling as two x derivatives.)

Explicit solutions can be obtained by inverse scattering (AxPgSac1997, FsSng1992, Zx1990), although this does not appear to be directly helpful to the low-regularity LWP and GWP theory.

The Cauchy problem is usually studied in two-parameter Sobolev spaces $H_{x}^{s_{1}}H_{y}^{s_{2}}$. However, the presence of inverse derivatives such as $\nabla _{x}^{-1}$ necessitates some technical modifications to these spaces at low frequencies which we will not detail here.

Despite the apparent similarity of KP-I and KP-II, the two equations behave very differently, both qualitatively and quantitatively. KP-I has the advantage of having a Hamiltonian which is positive definite in the leading order terms, however it contains resonances which make an X^{s,b}-based analysis somewhat delicate. The KP-II equation does not have any non-trivial resonances, but its Hamiltonian does not have a definite sign.

Just as KdV can be generalized to gKDV-k, KP can be generalized to gKP-k, by replacing $uu_{x}$ with $u^{k}u_{x}$ (and $u^{3}/3$ with $u^{k+1}/(k+1)$ in the Hamiltonian). The value $k=4/3$ is critical in the sense that the potential energy term in the Hamiltonian can be controlled by the other two terms, and thus one expects blow up in general for long times when $k>4/3$. This is supported by numerics (BnDgKar1986, WgxAbSe1994, Wc1987).