Convergence to a traveling wave solution for the Burgers-FKPP equation

We study the large time behavior of solutions for the Burgers-FKPP equation. When the coefficient $\beta$ of the Burgers nonlinearity increases, it leads to a phase transition from pulled fronts to pushed fronts. By introducing a weighted Hopf-Cole transform, we capture the criticality of phase transitions at $\beta =2$. With that, we can show the convergence of a solution to a single traveling wave in the Burgers-FKPP equation for all $\beta$. I will further show how our new approach can improve the spreading speed results for the Keller-Segel-FKPP equation.

Close to the end of the talk, I will switch the topic and mention works in analyzing stochastic gradient algorithms from the continuous time limit perspective. I will discuss how such an approach can provide explanations of why two common bias-correcting methods in sampling, resampling and reweighting, can have different performance when stochastic gradient algorithms are applied.