In this talk, I will discuss a construction of “shock forming” solutions to a class of dispersive and dissipative perturbations of the Burgers equation. This class includes the fractional KdV equation with dispersive term of order $\alpha \in [0,1)$, the Whitham equation arising in water waves, and the fractal Burgers equation with dissipation term of order $\beta \in [0,1)$.

Our result seems to be the first construction of gradient blow-up for fractional KdV in the range $\alpha \in [2/3,1)$. We construct blow-up solutions by a self-similar approach, treating the dispersive term as perturbative.

The blow-up is stable for $\alpha < 2/3$. However, for $\alpha \geq 2/3$, the solution is constructed by perturbing an underlying unstable self-similar Burgers profile. The construction is carried out by means of a weighted $L^2$ approach, which may be of independent interest.

This is joint work with Sung-Jin Oh (UC Berkeley).

Zoom link: https://princeton.zoom.us/j/92147928280?pwd=aGJ4VStpUTI2RWh1Y2FqTjlGQnZGQT09