Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity

Abeya, Asela; Biondini, Gino; Prinari, B.

doi:10.4208/eajam.261021.230122

Cited by 5 publications

(3 citation statements)

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“…In fact, the problem that the IST for the defocusing Manakov system with nonparallel NZBC has been studied recently in Ref. 50, which is related to some physical phenomenon. And the study has shown that no reflectionless potentials can exist in the case of nonparallel NZBC, that is, the problem has no soliton solutions.…”

Section: A C K N O W L E D G M E N T Smentioning

confidence: 99%

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

Zhu

2023

Stud Appl Math

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The two-component Gerdjikov-Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann-Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.

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Section: A C K N O W L E D G M E N T Smentioning

confidence: 99%

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

Zhu

2023

Stud Appl Math

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“…We anticipate that an interesting direction for further study could be the effort to extend the results of the present work to the case of non-parallel BCs, i.e., the case when q † + q − < q ± 2 . The IST for the IVP with non-parallel BCs was recently presented in [1]. There are several complications arising in that case, however.…”

Section: Final Remarksmentioning

confidence: 99%

“…We start from the reconstruction formula (2.26) in the reflectionless case, in which case the integral term is absent. First of all we define the following quantities: d (1) n, j = e i(θ 2 (z j )−θ 1 (z j )) , (A.31c) d (7) n, j =…”

Section: A4 Determinantal Formula For the Multi-soliton Solutions Of ...mentioning

confidence: 99%

Manakov system with parity symmetry on nonzero background and associated boundary value problems

Abeya¹,

Biondini²,

Prinari³

2022

J. Phys. A: Math. Theor.

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We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-deﬁned spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reﬂection coefﬁcients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark-bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x) and the “mixed parity” cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.

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On Maxwell-Bloch Systems with Inhomogeneous Broadening and One-sided Nonzero Background

Abeya,

Biondini,

Kovačič

et al. 2024

Commun. Math. Phys.

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Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity

Cited by 5 publications

References 0 publications

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

Manakov system with parity symmetry on nonzero background and associated boundary value problems

On Maxwell-Bloch Systems with Inhomogeneous Broadening and One-sided Nonzero Background

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