2025/05/04 by Cade Ballew, Ballew, Cade, Deniz Bilman +3 · 1 voice
Mathematics · Physics and Astronomy · #35C08 #35Q15 #35Q53 #37K10 #37K15 #65E05 #65M99 #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Pattern Formation and Solitons (nlin.PS) #math-ph #math.NA #nlin.PS #nlin.SI
paper · pdf · doi:10.48550/arxiv.2505.02029
We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg--de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to -∞. This accumulation results in an associated Riemann--Hilbert problem on a number of disjoint intervals. In the case where the jump matrices have specific square-root behavior, we describe an efficient and accurate numerical method to solve this Riemann--Hilbert problem and extract the potential. The keys to the method are, first, the deformation of the Riemann--Hilbert problem, making numerical use of the so-called g-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation.