The Pavlov equation

\begin{align*}

v_{xt} + v_{yy} + v_x v_{xy} - v_y v_{xx} = 0,

v=v(x,y,t) \in \R, \, x,y,t \in \R

\end{align*}

is a toy model of the dispersionless Kadomtsev-Petviashvili (or Zabolotskaya-Khokhlov) equation. It is a simpler integrable model by noting that the Lax pair continas no derivatives with respect to the spectral parameter $\lambda$.

By a change of variables, the Pavlov equation turns into a a quasilinear wave equation satisfying the null condition of which the Cauchy problem with initial data on a non-characteristic hypersurface is solved gloablly with small data constraints by S.\ Alinhac. For large initial data, he studied the lifespan and justified blowups of cups type occur if some conditions are fulfilled.

Our studies are devoted to the above $2+1$ quasilinear wave equation with characteristic initial surfaces. Namely, using the inverse scattering method, we prove a global solvability of the Cauchy problem for the Pavlov equation if the initial data is small. For large initial data, we prove a local solvability and show that either a bifurcation or a gradient catastrophe must occur if the Pavlov equation cannot be solved globally.

This report is based on papers

[1.] P.\ G.\ Grinevich, P.\ M.\ Santini, and D.\ Wu: The Cauchy Problem for the Pavlove equation (Nonlinearity (2015), v. 26, no. 11, pp. 3709-3754)

[2.] D.\ Wu. The Cauchy problem for the Pavlov equation with large data (Journal of Differential Equations (2017), v. 263, Issue 3, 5 August, pp. 1874--1906)