Conjecture on gradient catastrophe and dispersive scaling for two-dimensional Toda equations

Consider initial data that are the $x$-derivative of a rapidly decreasing smooth function in $L_{2}(\mathbb{R}^{2})$ with a single maximum. Write the two-dimensional dispersionless Toda equation as

$$\rho u_{tt}=(e^{u})_{xx}+u_{yy},\qquad \rho=\pm1,$$

and let $t_{c}$ be the first critical time. The cases $\rho=1$ and $\rho=-1$ are called the hyperbolic and elliptic cases, respectively; let $\epsilon$ be the dispersion parameter and $t^{*}$ the elliptic blow-up time. Two-dimensional Toda conjecture. The following assertions are expected: (i) in the hyperbolic case there are one or more finite-time gradient-catastrophe points, generically cubic singularities with finite $L_{\infty}$ norm and singularity in only one direction in the $(x,y)$-plane; (ii) in the elliptic case there is a finite-time gradient-catastrophe point, generically a square-root singularity with finite $L_{\infty}$ norm and singularity in only one direction; (iii) at $t_{c}$, the hyperbolic and elliptic differences between dispersive and dispersionless solutions scale as $\epsilon^{2/7}$ and $\epsilon^{2/5}$, respectively; and (iv) for $\epsilon\ll1$, the elliptic solution blows up at a finite $t^{*}>t_{c}$, with $t^{*}-t_{c}$ tending to zero as $\epsilon^{0.9}$ as $\epsilon\to0$. These are numerical predictions for the two-dimensional Toda equations, motivated in part by analogous behavior for dispersionless Kadomtsev–Petviashvili equations; their general validity remains open.