Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schr�dinger problem and the KPI equation

Boiti, M.; Pempinelli, F.; Pogrebkov, A. K.; Polivanov, M. C.

doi:10.1007/bf01083519

Cited by 28 publications

(32 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, let D j denote the extension of the differential operator ∂ xj (j = 1, 2), i.e. according to (2.8) 9) then D j in the (p, q)-representation takes the form…”

Section: Direct and Inverse Problems In The Case Of Rapidly Decaying mentioning

confidence: 99%

Towards an inverse scattering theory for non-decaying potentials of the heat equation

et al. 2001

View full text Add to dashboard Cite

Abstract. The resolvent approach is applied to the spectral analysis of the heat equation with non decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe N solitons superimposed by Bäcklund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a ∂-problem explicitly in terms of the corresponding objects associated to the original potential. Regularity conditions of the potential in the cases N = 1 and N = 2 are investigated in details. The singularities of the resolvent for the case N = 1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.

show abstract

“…In particular, let D j denote the extension of the differential operator ∂ xj (j = 1, 2), i.e. according to (2.8) 9) then D j in the (p, q)-representation takes the form…”

Section: Direct and Inverse Problems In The Case Of Rapidly Decaying mentioning

confidence: 99%

Towards an inverse scattering theory for non-decaying potentials of the heat equation

et al. 2001

View full text Add to dashboard Cite

show abstract

“…The operators on this set of functions are (infinite) matrices F m,m , where m and m are double indices, m = (m 1 , m 2 ), m = (m 1 , m 2 ), and m 1 , m 2 , m 1 , m 2 ∈ Z. In order to develop our construction, we embed a subspace of these matrices in the set of some special objects; by analogy with the continuous case, these objects will be called extended operators; see [3]- [6].…”

Section: Commutator Identity and A Linear Difference Equationmentioning

confidence: 99%

Hirota difference equation and a commutator identity on an associative algebra

Pogrebkov

2011

St. Petersburg Math. J.

Self Cite

View full text Add to dashboard Cite

Abstract. In earlier papers of the author it was shown that some simple commutator identities on an associative algebra generate integrable nonlinear equations. Here, this observation is generalized to the case of difference nonlinear equations. The identity under study leads, under a special realization of the elements of the associative algebra, to the famous Hirota difference equation. Direct and inverse problems are considered for this equation, as well as some properties of its solutions. Finally, some other commutator identities are discussed and their relationship with integrable nonlinear equations, both differential and difference, is demonstrated. §1. Algebraic identities 1

show abstract

“…Say, identity associated to the Kadomtsev-Petviashvili equation has the form We also suggested there representation of elements A and B as extended operators in terms of the approach developed in [3]- [7], that can be considered as a version of the dressing procedure. Then we proved that under some general conditions imposed on this representation one can uniquely reconstruct both, nonlinear equations themselves and their Lax pairs, once the commutator identities are given.…”

Section: Introduction Commutator Identity and Linearized Version Of mentioning

confidence: 99%

“…In Sec. 2 we introduce mathematical details of this construction based on the method of the extended resolvent, see [3]- [7]. In Sec.…”

Section: Introduction Commutator Identity and Linearized Version Of mentioning

confidence: 99%

2D Toda chain and associated commutator identity

Pogrebkov¹

2008

Geometry, Topology, and Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

Resolvent approach for two-dimensional scattering problems. Application to the nonstationary Schr�dinger problem and the KPI equation

Cited by 28 publications

References 4 publications

Towards an inverse scattering theory for non-decaying potentials of the heat equation

Towards an inverse scattering theory for non-decaying potentials of the heat equation

Hirota difference equation and a commutator identity on an associative algebra

2D Toda chain and associated commutator identity

Contact Info

Product

Resources

About