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Abstract In the KdV context, we revisit the classical Darboux transformation in the framework of the vector Riemann–Hilbert problem. This readily yields a version of the binary Darboux transformation providing a short‐cut to explicit formulas for solitons traveling on a wide range of background solutions. Our approach also links the binary Darboux transformation and the double commutation method.
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A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation
Abstract In the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step‐type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.
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- Award ID(s):
- 2009980
- PAR ID:
- 10419877
- Publisher / Repository:
- Wiley-Blackwell
- Date Published:
- Journal Name:
- Studies in Applied Mathematics
- Volume:
- 151
- Issue:
- 1
- ISSN:
- 0022-2526
- Page Range / eLocation ID:
- p. 208-246
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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