Darboux-type transformations and hyperelliptic curves

Gesztesy, Fritz; Holden, Helge

doi:10.1515/crll.2000.080

Cited by 7 publications

(11 citation statements)

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“…They have been discussed, for instance, in [27] and [28]. In the special context of algebro-geometric AKNS solutions, the effect of elementary Darboux transformations on the underlying compact hyperelliptic curve (in connection with the insertion and deletion of eigenvalues as well as the isospectral case) was studied in detail in [15], [16, App. G] (see also [19], [20]).…”

Section: Darboux-type Transformations For Akns and Nls − Systemsmentioning

confidence: 99%

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

et al. 2004

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Abstract. We study Darboux-type transformations associated with the focusing nonlinear Schrödinger equation (NLS − ) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J -selfadjoint (but non-self-adjoint) Dirac-type differential expression of the formAs one of our principal results we prove that under the most general hypothesis q ∈ L 1 loc (R) on q, the maximally defined operatorThe Darboux transformations considered in this paper are the analog of the double commutation procedure familiar in the KdV and Schrödinger operator contexts. As in the corresponding case of Schrödinger operators, the Darboux transformations in question guarantee that the resulting potentials q are locally nonsingular. Moreover, we prove that the construction of Nsoliton NLS − potentials q (N) with respect to a general NLS − background potential q ∈ L 1 loc (R), associated with the Dirac-type operators D q (N) and D(q), respectively, amounts to the insertion of N complex conjugate pairs of L 2 (R) 2 -eigenvalues {z 1 , z 1 , . . . , z N , z N } into the spectrum σ(D(q)) of D(q), leaving the rest of the spectrum (especially, the essential spectrum σe(D(q))) invariant, that is,These results are obtained by establishing the existence of bounded transformation operators which intertwine the background Dirac operator D(q) and the Dirac operator D q (N) obtained after N Darboux-type transformations.

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Section: Darboux-type Transformations For Akns and Nls − Systemsmentioning

confidence: 99%

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

et al. 2004

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“…That q in (3.1), (3.2), and (3.4) are all isospectral KdV potentials, that is, they all belong to the same algebraic curve (3.5) (assuming (3.1) and (3.2) satisfy the additional restrictions to make them algebrogeometric KdV potentials, of course), can be shown by several methods, either by invoking time-dependent KdV flows as in [5], or by commutation techniques (i.e., Darboux-type transformations) as in [3,11,30,31] (cf. also [14]). This fact also follows from the results in [38].…”

Section: Hence a Comparison With (A12)-(a15) (A19)-(a24) Yieldsmentioning

confidence: 87%

On a Theorem of Halphen and its Application to Integrable Systems

Gesztesy

Unterkofler

Weikard

2000

Journal of Mathematical Analysis and Applications

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We extend Halphen's theorem which characterizes solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n × n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with completely integrable models of the Calogero-Moser type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles.

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“…Then the (isospectral) Dirichlet deformation (µ, σ) → (µ, −σ) is precisely the isospectral case of the double commutation method (cf. [74], [78], [100, App. B], [103]).…”

Section: )mentioning

confidence: 99%