Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

Valchev, Tihomir I.

doi:10.1063/1.4940996

Cited by 6 publications

(6 citation statements)

References 30 publications

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“…One of the most convenient approaches to the derivation of the soliton solutions is the so-called dressing method [74,75] (see also [23,36,73,40,70,26]). The rationale of the method is the construction of a nontrivial (dressed) FAS, χ ± (x, t, λ) from the known (bare) FAS, χ ± 0 (x, t, λ) by the means of the so-called dressing factor u(x, t, λ) :…”

Section: Dressing Methodsmentioning

confidence: 99%

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On integrable wave interactions and Lax pairs on symmetric spaces

Gerdjikov

Grahovski

Ivanov³

2017

Wave Motion

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Multi-component generalizations of derivative nonlinear Schrodinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z_2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann-Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup-Newell (KN) and Gerdjikov-Ivanov (GI) types is derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundamental properties of the multi-component GI equations are briefly discussed at the end.Comment: 27 pages, LaTe

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Section: Dressing Methodsmentioning

confidence: 99%

“…Later on, this approach was extended to other types of multi-component integrable models, like the derivative NLS, Korteweg-de Vries and modified Korteweg-de Vries, N-wave, Davey-Stewartson, Kadomtsev-Petviashvili equations [6,17]. For example, the equation [17] (see also [68,70])…”

Section: Introductionmentioning

confidence: 99%

On integrable wave interactions and Lax pairs on symmetric spaces

Gerdjikov

Grahovski

Ivanov³

2017

Wave Motion

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“…The RH method is a very efficient method for obtaining soliton solutions. We obtained some results in this paper and we look forward to studying multicomponent systems, which include different Lie algebras [38], quadratic spectral parameters [21,39,40], and polynomial spectral parameters [41]. In addition, we will investigate where the jump matrix can be obtained at non-zero boundary branch cuts and use the RH method to obtain rogue waves.…”

Section: Discussionmentioning

confidence: 99%

“…K + is analytic in Γ + and continuous to Γ 0 . Substituting K + into Equation (40), it is evident that…”

Section: Riemann-hilbert Problemmentioning

confidence: 99%

The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution

2022

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It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures.

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“…Here, we consider multicomponent derivative NLS-type (DMNLS) equations [35,77] and multicomponent GI (MGI) equations [52,53]. Note that the RHP for the DMNLS equations is not canonically normalized which requires slight modifications in applying the dressing method.…”

Section: Lax Pairs On Symmetric Spaces Generic Casementioning

confidence: 99%

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

Gerdjikov,

Stefanov

2023

Symmetry

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The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: (1) Lax representation [L,M]=0; (2) construction of fundamental analytic solutions (FAS); (3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) ξ+(x,t,λ)=ξ−(x,t,λ)G(x,tλ) on a contour Γ with sewing function G(x,t,λ); (4) soliton solutions and possible applications. Step 1 involves several assumptions: the choice of the Lie algebra g underlying L, as well as its dependence on the spectral parameter, typically linear or quadratic in λ. In the present paper, we propose another approach that substantially extends the classes of integrable NLEE. Its first advantage is that one can effectively use any polynomial dependence in both L and M. We use the following steps: (A) Start with canonically normalized RHP with predefined contour Γ; (B) Specify the x and t dependence of the sewing function defined on Γ; (C) Introduce convenient parametrization for the solutions ξ±(x,t,λ) of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); (D) use Zakharov–Shabat dressing method to derive their soliton solutions. This requires correctly taking into account the symmetries of the RHP. (E) Define the resolvent of the Lax operator and use it to analyze its spectral properties.

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Dressing method and quadratic bundles related to symmetric spaces. Vanishing boundary conditions

Cited by 6 publications

References 30 publications

On integrable wave interactions and Lax pairs on symmetric spaces

On integrable wave interactions and Lax pairs on symmetric spaces

The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution

Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions

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