General N-soliton solution of the AKNS class on arbitrary background

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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“…4.2], [36], [41]) one introduces the quantities ϕ k (x) = ψ 2,− (z k , x) + γ k ψ 2,+ (z k , x) ψ 1,− (z k , x) + γ k ψ 1,+ (z k , x) , …”

Section: Transformation Operators For J -Self-adjoint Dirac-type Operunclassified

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

et al. 2004

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“…, Q 2N expressed in the similar way. Our method is closely related to the Neugebauer-Meinel approach [3]. Let D is given by (38).…”

Section: The Multi-soliton Darboux Matrixmentioning

confidence: 99%

“…There are several methods to construct the Darboux matrix (which generates soliton solutions) [1,2,3,4,6,5,7,8]). However, these methods are technically difficult when applied to the matrix versions of the spectral problems which are naturally represented in Clifford algebras [9,10,12].…”

Section: Introductionmentioning

confidence: 99%

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A new approach to the Darboux–Bäcklund transformation versus the standard dressing method

Cieśliński¹,

Biernacki²

2005

J. Phys. A: Math. Gen.

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We present a new approach to the construction of the Darboux matrix. This is a generalization of the recently formulated method based on the assumption that the square of the Darboux matrix vanishes for some values of the spectral parameter. We consider the multisoliton case, the reduction problem and the discrete case. The relationship between our approach and the standard dressing method is discussed in detail.

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“…Through solving the relevant linear system with a given seed solution, with no need to refer to the special boundary conditions, a series of new analytical solutions can be generated under the Darboux transformation. By performing the iterative algorithm of the Darboux transformation successively, one can obtain the n-times iterated potential transformation formula in terms of the Wronskian determinant [22] or Vandermonde-like determinant [26]. This paper is devoted to applying the Darboux transformation method to System (3) based on the Lax pair derived from the matrix AKNS scheme.…”

Section: Introductionmentioning

confidence: 99%

Darboux Transformation and Symbolic Computation on Multi-Soliton and Periodic Solutions for Multi-Component Nonlinear Schr¨odinger Equations in an Isotropic Medium

Zhang

et al. 2009

Zeitschrift Für Naturforschung A

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The Darboux transformation is applied to a multi-component nonlinear Schrödinger system, which governs the propagation of polarized optical waves in an isotropic medium. Based on the Lax pair associated with this integrable system, the formula for the n-times iterative Darboux transformation is constructed in the form of block matrices. The purely algebraic iterative algorithm is carried out via symbolic computation, and two different kinds of solutions of practical interest, i. e., bright multi-soliton solutions and periodic solutions, are also presented according to the zero and nonzero backgrounds.

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General N-soliton solution of the AKNS class on arbitrary background

Cited by 194 publications

References 4 publications

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

Spectral analysis of Darboux transformations for the focusing NLS hierarchy

A new approach to the Darboux–Bäcklund transformation versus the standard dressing method

Darboux Transformation and Symbolic Computation on Multi-Soliton and Periodic Solutions for Multi-Component Nonlinear Schr¨odinger Equations in an Isotropic Medium

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