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Abstract: One of the difficulties associated with the scattering problems arising inconnection with integrable systems is that they are frequentlynon-self-adjoint, making it difficult to determine where the spectrum lies. Inthis paper, we consider the problem of locating and counting the discreteeigenvalues associated with the scattering problem for which the sine-Gordonequation is the isospectral flow. In particular, suppose that we take aninitially stationary pulse for the sine-Gordon equation, with a profile thathas either one extremum point of height less than pi and topological charge 0,or is monotone with topological charge +-1. Then we show that the pointspectrum lies on the unit circle and is simple. Furthermore, we give a count ofthe number of eigenvalues. This result is an analog of that of Klaus and Shawfor the Zakharov-Shabat scattering problem. We also relate our results, as wellas those of Klaus and Shaw, to the Krein stability theory for symplecticmatrices. In particular we show that the scattering problem associated to thesine-Gordon equation has a symplectic structure, and under the above conditionsthe point eigenvalues have a definite Krein signature, and are thus simple andlie on the unit circle.



Author: Jared C. Bronski, Mathew A. Johnson

Source: https://arxiv.org/



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