New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

Tsuchida, Takayuki

doi:10.1063/1.3315862

Cited by 27 publications

(36 citation statements)

References 52 publications

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“…while the solutions of (24) contains an extra derivative (57). This explains why the solutions of (26) are connected to those of (35) through the potential variable ψ = ∂ x ϕ [27,31]. Comparing (59) and (60) with the gauge transformation (38), we note that the form of the transformation connecting the three types of DNLS equations are already contained in the dressing operators.…”

Section: Dressing Approach To the Kn Hierarchymentioning

confidence: 93%

The algebraic structure behind the derivative nonlinear Schrödinger equation

França¹,

Gomes²,

Zímerman³

2013

J. Phys. A: Math. Theor.

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The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schrödinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of aŝ 2 Kac-Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows, respectively. The equivalence between the latter and the massive Thirring model is explicitly demonstrated also. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.

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Section: Dressing Approach To the Kn Hierarchymentioning

confidence: 93%

The algebraic structure behind the derivative nonlinear Schrödinger equation

França¹,

Gomes²,

Zímerman³

2013

J. Phys. A: Math. Theor.

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“…This equation is the potential form of an equation given with its Lax pairs by Tsuchida in [25]. (17).…”

Section: Systemmentioning

confidence: 99%

Two-component integrable generalizations of Burgers equations with nondiagonal linearity

Talati

Turhan

2016

Journal of Mathematical Physics

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“…which is known as the semi-discrete derivative NLS system or the semi-discrete Kaup-Newell system. [1][2][3] From the denominators in (3), we see why we need the restriction (2). Note that an overdot in (3) denotes the derivative with respect to the independent variable 𝑡, which is interpreted as the time variable and is suppressed in (3) by writing 𝑞 𝑛 and 𝑟 𝑛 instead of 𝑞 𝑛 (𝑡) and 𝑟 𝑛 (𝑡), respectively.…”

Section: Introductionmentioning

confidence: 99%

Direct and inverse scattering problems for the first‐order discrete system associated with the derivative NLS system

Aktosun

Ercan

2021

Stud Appl Math

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The direct and inverse scattering problems are analyzed for a first-order discrete system associated with the semidiscrete version of the derivative nonlinear Schrödinger (NLS) system. The Jost solutions, the scattering coefficients, the bound-state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi-discrete version of the NLS system. The bound-state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to solve the inverse problem to recover the potential values in the first-order discrete system. One of these methods uses a newly derived, standard discrete Marchenko system using as input the scattering data directly coming from the first-order discrete system. This new Marchenko method is presented in a way that it is generalizable to other first-order systems both in the discrete and continuous cases for which a Marchenko system and a Marchenko theory are not yet available. Finally, using the time-evolved scattering data set, the inverse scattering transform is applied on the corresponding semi-discrete derivative NLS system, and in

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New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

Cited by 27 publications

References 52 publications

The algebraic structure behind the derivative nonlinear Schrödinger equation

The algebraic structure behind the derivative nonlinear Schrödinger equation

Two-component integrable generalizations of Burgers equations with nondiagonal linearity

Direct and inverse scattering problems for the first‐order discrete system associated with the derivative NLS system

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