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I have a question about two well-known spectral problems in Integrable Systems. These are the Dirac and the ZS-AKNS spectral problems. They are are known to be gauge equivalent (please see equations (3) and (5) in the paper: http://ac.els-cdn.com/S0375960199007665/1-s2.0-S0375960199007665-main.pdf?_tid=84c439b4-185f-11e6-9d95-00000aab0f27&acdnat=1463071101_131dcf6fe094a2dd3bd0a7458d1e12ac)

Obviously, the existence of the gauge transformation between them indicates that they generate the same soliton hierarchy, so my question is: why do these two spectral problems both exist if they are gauge equivalent? In other words, why have the two been studied separately for many years? Is there any advantage of one over the other? Are there any known differences?

By the way I just started doing some research in this field and I would be grateful if anyone can give me their expert opinion on this issue.

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  • $\begingroup$ you will perhaps want to be a bit more explicit on what you would like to know, for a more helpful response; as far as I know, this is just one single spectral problem that goes by different names in different contexts; I'm not sure there is more to say. $\endgroup$ May 19, 2016 at 9:47
  • $\begingroup$ @CarloBeenakker. If you look at the reference I gave, the author indicates that the spectral problems are different but mentions that they are gauge equivalent. He further states that their extensions have different r-matrices that are nondynamical (i.e. do not depend on the dynamical variables). You should note that despite such a difference, their extensions are also gauge equivalent. So I do not think it is valid to say they are the same with different names. I think there should be some differences which is what I am looking for. $\endgroup$
    – smanoos
    May 19, 2016 at 18:54

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