Darboux dressing and undressing for the ultradiscrete KdV equation

Nimmo, J. J.C., Gilson, C. R. and Willox, R. (2019) Darboux dressing and undressing for the ultradiscrete KdV equation. Journal of Physics A: Mathematical and Theoretical, 52(44), 445201. (doi: 10.1088/1751-8121/ab45cf)

Abstract

We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We then show how to reconstruct the potential in the scattering problem at any time, using an ultradiscrete analogue of a Darboux transformation. This is achieved in two steps. First we use so-called 'undressing' transformations (i.e. Darboux transformations that use bound state eigenfunctions) on the potential in the scattering problem to obtain data that uniquely characterises both its soliton content and the remaining 'background', and thereby the entire potential. Then we use so-called 'dressing' transformations (i.e. Darboux transformations that use bound state eigenfunctions) to put back the solitonic content in the background at general time t, whereby obtaining an explicit expression for the solution to the udKdV equation associated to the original potential at t  =  0.

• Item Type: Articles
• Additional Information: JJCN would like to acknowledge partial support from the Edinburgh Mathematical Society Research Fund and from the Glasgow Mathematical Journal Trust. RW would like to acknowledge support from the Japan Society for the Promotion of Science (JSPS), through the JSPS grant: KAKENHI grant number 15K04893. He would also like to express his gratitude for financial support (from EMS and GMJT) during a visit to Glasgow in spring 2016, during which this work took shape.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Gilson, Dr Claire and Nimmo, Dr Jonathan
• Authors: Nimmo, J. J.C., Gilson, C. R., and Willox, R.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Physics A: Mathematical and Theoretical
• Publisher: IOP Publishing
• ISSN: 1751-8113
• ISSN (Online): 1751-8121
• Published Online: 09 October 2019
• Copyright Holders: Copyright © 2019 IOP Publishing Ltd
• First Published: First published in Journal of Physics A: Mathematical and Theoretical 52(44):445201
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher