KP-II equation with upper semicontinuous initial data

Problem's Description

In the article [1], they study the KP-II equation: the function $\phi(t,x,r)=\partial_{r}^{2}\log(F)$ satisfies

$$\partial_{t}\phi+\frac{1}{2}\partial_{r}(\phi)^{2}+\frac{1}{12}\partial_{r}^{3}(\phi)+\frac{1}{4}\partial_{r}^{-1}\partial_{x}^{2}\phi=0$$

or in the Hirota-form

$$F\partial_{tr}^{2}F-\partial_{t}F\partial_{r}F+\frac{1}{12}F\partial_{r}^{4}F-\frac{1}{3}\partial_{r}F\partial_{r}^{3}F+\frac{1}{4}(\partial_{r}^{2}F)^{2}+\frac{1}{4}F\partial_{x}^{2}F-\frac{1}{4}(\partial_{x}F)^{2}=0$$

with the following initial data (see example 1.6): first consider the space of upper semicontinuous functions

$$UC=\{h:\mathbb{R}\to [-\infty,\infty): \text{ h is upper semicontinuous}, h(x)\leq A+B|x|, A,B>0, h\not\equiv -\infty\},$$ we then fix some $h_{0}(x)\in UC$ and set

$$\phi(0,x,r):=0,\text{ for }r\geq h_{0}(x),~~~~~,\phi(0,x,r):=-\infty,\text{ for }r< h_{0}(x).$$

The research problem is to prove well-posedness of this initial value problem.

1. Conference PaperIs an originKP governs random growth off a 1-dimensional substrate

   • Jeremy Quastel,
   • Daniel Remenik

   pp. e10, year of publication: 2022