Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy

Gomes, J. F.; França, Guilherme; Zímerman, A. H.

doi:10.1088/1742-6596/343/1/012039

Cited by 3 publications

(6 citation statements)

References 13 publications

(40 reference statements)

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“…The DNLSE-I (26) arises from the second positive flow, while the Mikhailov model (35) is obtained from the first negative flow. The gauge transformation (38) connects the system (24) to (39), which in particular yields relations between the three kinds of DNLS equations, namely, (26), (40) and (41). Furthermore, we have demonstrated a general relation between the model (35) and the massive Thirring model (45) through (48).…”

Section: Discussionmentioning

confidence: 75%

“…where A = 8c (2c − 1), B = 4c and C = 4 (1 − c). Note that in (39) we have a fifth-order nonlinearity and two types of derivative nonlinear terms. Let us use the same transformation leading to equation ( 26), i.e.…”

Section: Relations Between Dnls Type Equationsmentioning

confidence: 99%

“…This algebraic approach is well understood for models that fit into the AKNS construction, where the fields are associated with zero grade operators. Nevertheless, there are some generalizations of this formalism, for instance, the addition of fermionic fields to higher grading operators [37,38] and a generalization of the dressing method to include NVBC for the AKNS construction [39][40][41]. Recently, a higher grading construction was proposed that includes the Wadati-Konno-Ichikawa hierarchy, and the algebraic dressing formalism was supplemented with reciprocal transformations [42].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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: Discussionmentioning

confidence: 75%

Section: Relations Between Dnls Type Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The algebraic structure behind the derivative nonlinear Schrödinger equation

França¹,

Gomes²,

Zímerman³

2013

J. Phys. A: Math. Theor.

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“…The fields in B are directly related to the famous tau functions, obtained by projecting (6.2) between the highest weight states of the algebra (for more details we refer the reader to [24,25]). The adjoint relations are E j…”

Section: Akns Solutionsmentioning

confidence: 99%

The higher grading structure of the WKI hierarchy and the two-component short pulse equation

França

Gomes

Zímerman

2012

J. High Energ. Phys.

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A higher grading affine algebraic construction of integrable hierarchies, containing the Wadati-Konno-Ichikawa (WKI) hierarchy as a particular case, is proposed. We show that a two-component generalization of the Schäfer-Wayne short pulse equation arises quite naturally from the first negative flow of the WKI hierarchy. Some novel integrable nonautonomous models are also proposed. The conserved charges, both local and nonlocal, are obtained from the Riccati form of the spectral problem. The loop-soliton solutions of the WKI hierarchy are systematically constructed through gauge followed by reciprocal Bäcklund transformation, establishing the precise connection between the whole WKI and AKNS hierarchies. The connection between the short pulse equation with the sine-Gordon model is extended to a correspondence between the two-component short pulse equation and the Lund-Regge model. *

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“…It has been shown that the AKNS equations are completely integrable. The researches about AKNS equations are in progress up to now [9][10][11]. In this paper, we will present a nonlinear evolution equation on the basis of the AKNS equations.…”

Section: Introductionmentioning

confidence: 99%