Riemann-Hilbert Problems, families of commuting operators and soliton equations

Gerdjikov, Vladimir S.; Yanovski, A.B.

doi:10.1088/1742-6596/482/1/012017

Cited by 9 publications

(13 citation statements)

References 14 publications

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“…We have also formulated the Riemann-Hilbert problem for the Kaup-Newell (KN) and GI equations on A.III-symmetric spaces and derived explicit parametrization of the associated Lax operators. This can serve as a starting point in obtaining the Lax pair and the corresponding NLEEs [24,37]. Finally, we have presented a modification of the dressing method and obtained 1-solitons for the multi-component GI equation with local and nonlocal reductions.…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…Let us detail the consequences of the reductions 1) and 3) in (37) on the NLEE (20). It is easy to see that they restrict U 0 (x, t) and Q(x, t) by: 1)…”

Section: Local Reductionsmentioning

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: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

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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“…These equations were derived directly from a ZCR that is quadratic in a spectral parameter, and were shown to admit solutions that may be constructed via the Riemann-Hilbert problem. Polynomial dependence of ZCRs on a parameter was also discussed in [10].…”

Section: Introductionmentioning

confidence: 99%

IntegrableG-strands on semisimple Lie groups

Gay–Balmaz¹,

Holm²,

Raţiu³

2014

J. Phys. A: Math. Theor.

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“…Their Lax representations in fact are closely related to the Kac-Moody algebras [15]. The spectral properties of their Lax operators [29,28,30] however are not completely established and deserve further studies.…”

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confidence: 99%

Riemann-Hilbert problem, integrability and reductions

Gerdjikov¹,

Ivanov²,

Stefanov³

2019

Journal of Geometric Mechanics

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The present paper is dedicated to integrable models with Mikhailov reduction groups G R ≃ D h . Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the G R -action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with D h symmetries are presented. 2010 Mathematics Subject Classification. Primary: 35K10, 35Q15; Secondary: 37K15, 35Q55.

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Riemann-Hilbert Problems, families of commuting operators and soliton equations

Cited by 9 publications

References 14 publications

On integrable wave interactions and Lax pairs on symmetric spaces

On integrable wave interactions and Lax pairs on symmetric spaces

IntegrableG-strands on semisimple Lie groups

Riemann-Hilbert problem, integrability and reductions

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