Long-time asymptotics of the KdV equation with delta function initial profile

This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent method. When the initial value is a delta potential well, the asymptotic solution is predominantly dominated by a single soliton in certain region for $x>0$, while in other regions, the dispersive tails including self-similar region, collisionless shock region and dispersive wave region, play a more significant role. Conversely, when the initial value is a delta potential barrier, the soliton region is absent, although the dispersive tails still persist. Moreover, the general delta function initial profile with $L$-spikes is also studied and it is proved that one to $L$ solitons will be generated in soliton region, which depends on the sizes of the distance and height of the spikes. The leading-order terms of the solution in each region are derived, highlighting the efficacy of the Riemann-Hilbert formulation in elucidating the long-time behaviors of integrable systems.