Non-unique singular solutions for KP and modified KP equations on ${\mathbb{T}}^2$ and ${\mathbb{R}}^2$

Abstract

We construct infinitely many weak singular solutions with zero initial data and compact time support for third- and fifth-order KP-I, KP-II, and their modified counterparts on ${\mathbb{T}}^2$ and ${\mathbb{R}}^2$. Their nonlinearities are cutoff-independent, absolutely convergent Fourier convolutions. Quadratic solutions belong to $C_t{L}^p$ for $p<2$ and $C_t(H^{-\sigma ,0}\cap H^{-\sigma })$ for $\sigma >0$. Modified solutions belong to $C_t{L}^p$ for $p<3$. One cubic family lies in $C_t{H}^{\alpha }$ for $\alpha <1/3$; another has parabolic Fourier support and lies in $C_t{H}^{s,0}$ for $s<1/2$ and $C_t{H}^{\alpha }$ for $\alpha <1/4$. The exponents $1/3$ and $1/2$ are sharp at the $L^3$ product threshold. For quadratic fifth-order KP on ${\mathbb{R}}^2$, the nonuniqueness range is almost sharp. We also construct periodic stationary KP-I and KP-II solutions and prove that $L^2$ is the sharp threshold between singular stationary KP-I solutions and smoothness.

49 pages