2019/08/29 by Piorkowski, Mateusz
#35Q15 (Secondary) #35Q53 (Primary) 37K45 #37K40 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.11340
In this paper we study the asymptotics of solutions to the Korteweg--de Vries equation with steplike initial data, which lead to shock waves in the region between the asymptotically constant region and the soliton region, as t → ∞. To achieve this, we present an alternative approach to the usual argument involving a small norm Riemann--Hilbert problem, which is based instead on the direct comparison of resolvents related to the corresponding Riemann--Hilbert problems. The motivation for this approach stems from the fact that an invertible holomorphic global parametrix solution for our problem does not exist for certain discrete times.