On numerical inverse scattering for the Korteweg-de Vries equation with\n discontinuous step-like data

2018/09/24 by Deniz Bilman, Bilman, Deniz, Thomas Trogdon +1
Earth and Planetary Sciences · Mathematics · #33F05 #35Q53 #37K15 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Pattern Formation and Solitons (nlin.PS) #Seismic Imaging and Inversion Techniques #Seismic Waves and Analysis

paper · pdf · doi:10.48550/arxiv.1809.09263

openalex publication_date 2018/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a method to compute dispersive shock wave solutions of the\nKorteweg-de Vries equation that emerge from initial data with step-like\nboundary conditions at infinity. We derive two different Riemann-Hilbert\nproblems associated with the inverse scattering transform for the classical\nSchr "odinger operator with possibly discontinuous, step-like potentials and\ndevelop relevant theory to ensure unique solvability of these problems. We then\nnumerically implement the Deift-Zhou method of nonlinear steepest descent to\ncompute the solution of the Cauchy problem for small times and in two\nasymptotic regions. Our method applies to continuous and discontinuous data.\n

Related

• Parametrix problem for the Korteweg--de Vries equation with steplike initial data
• Inverse scattering transform of the coupled modified Korteweg-de Vries equation with nonzero boundary conditions
• On the large-time asymptotics of the defocusing mKdV equation with step-like initial data
• Asymptotics of KdV shock waves via the Riemann-Hilbert approach
• Soliton asymptotics for KdV shock waves via classical inverse scattering