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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 $$-\infty$$. This accumulation results in an associated Riemann-Hilbert Problem (RHP) on a number of disjoint intervals. In the case where the jump matrices have specific square-root behaviour, we describe an efficient and accurate numerical method to solve this RHP and extract the potential. The keys to the method are, first, the deformation of the RHP, 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.
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Analysis of an approximation to a fractional extension problem
The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results.
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- Award ID(s):
- 1903450
- PAR ID:
- 10149920
- Date Published:
- Journal Name:
- BIT Numerical Mathematics
- ISSN:
- 0006-3835
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10543-019-00787-y
