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

TL;DR

This paper analyzes the long-time behavior of solutions to the KdV equation with delta function initial profiles, revealing soliton formation and dispersive tail dynamics using Riemann-Hilbert techniques.

Contribution

It introduces a detailed asymptotic analysis for delta initial profiles, including multiple spikes, using Riemann-Hilbert methods, and characterizes soliton and dispersive regions.

Findings

Single soliton dominates for delta well in certain regions.

Dispersive tails are significant in other regions.

Multiple spikes generate multiple solitons depending on spike parameters.

Abstract

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.