The Fredholm determinant method for discrete integrable evolution equations

Bauhardt, Wolfgang; Pöppe, Christoph

doi:10.1007/bf00423443

Cited by 6 publications

(3 citation statements)

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“…For the theory of Fredholm determinants for integral operators the reader is referred to [16]; a more general and more recent treatment of infinite-dimensional determinants is given in [ 151. The special analytical problems for integral operators on the half-infinite interval are discussed in [ l l ] , and for infinite-matrix operators related to Toda lattice equations in [4].…”

Section: Introductionmentioning

confidence: 99%

General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy

Pöppe

1989

Inverse Problems

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The t function, introduced by the 'Kyoto school' as a central element in the description of soliton equation hierarchies, is identified with the determinant of a family of linear operators solving linear, constant-coefficient PDE in the hierarchy variables, for the Kadomtsev-Petviashvili (KP) hierarchy. For Gel'fand-Levitan-Marchenko integral operators, the t function is the Fredholm determinant; we give a new proof of this fact. We show further that the determinant of a suitably chosen family of finite-dimensional matrices is a t function which gives rise to rational solutions of the K P equation. Finally. we prove the 'vertex operator identity' for T functions in the Fredholm determinant case.

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Section: Introductionmentioning

confidence: 99%

General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy

Pöppe

1989

Inverse Problems

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“…The key results are known (cf. Gesztesy and Teschl [18] see also [1,3,7,8,11,34]), but our representation for the general solutions u N in (2.13) is new. This version is more convenient for our purposes; a derivation can be found in the Appendix.…”

Section: Introductionmentioning

confidence: 94%

Toda Soliton Limits on General Backgrounds

Renger

1999

Journal of Differential Equations

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Starting from an arbitrary background solution of the Toda lattice, we study limits of N-soliton solutions on this given background as N tends to infinity. This yields a new class of solutions of the Toda lattice. Simultaneously, we solve an inverse spectral problem for one dimensional Jacobi operators we explicitly construct Jacobi operators whose spectrum contains a given (countable, bounded) set of eigenvalues and whose absolutely continuous spectrum coincides with that of a given background operator. Academic Press

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“…Subsequently, the second author of the present paper was able to extend it in [23] to the discrete situation exemplified by the Toda lattice. Her treatment of the discrete situation generalizes work of Bauhardt and Pöppe [4] where Fredholm determinants are used to solve the bilinear equation of Hirota which corresponds to the Toda lattice (see [14]). Moreover, in contrast to the work of Bauhardt and Pöppe, it is exactly the same formalism working both in the continuous as well as in the discrete situation.…”

Section: Introductionmentioning

confidence: 97%

Nonlinear equations in soliton physics and operator ideals

Carl¹,

Schiebold²

1999

Nonlinearity

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An operator-theoretic method for the investigation of nonlinear equations in soliton physics is discussed comprehensively.Originating from pioneering work of Marchenko, our operator-method is based on new insights into the theory of traces and determinants on operator ideals. Therefore, we give a systematic and concise approach to some recent developments in this direction which are important in the context of this paper.Our method is widely applicable. We carry out the corresponding arguments in detail for the Kadomtsev-Petviashvili equation and summarize the results concerning the Korteweg-de Vries and the modified Korteweg-de Vries equation as well as for the sine-Gordon equation.Exactly the same formalism works in the discrete case, as the treatment of the Toda lattice, the Langmuir and the Wadati lattice shows.

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The Fredholm determinant method for discrete integrable evolution equations

Cited by 6 publications

References 6 publications

General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy

General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy

Toda Soliton Limits on General Backgrounds

Nonlinear equations in soliton physics and operator ideals

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