Abstract:
We study a matrix Riemann–Hilbert (RH) problem for the modified Landau–Lifshitz (mLL) equation with nonzero boundary conditions at infinity. In contrast to the case of zero boundary conditions, multivalued functions arise during direct scattering. To formulate the RH problem, we introduce an affine transformation converting the Riemann surface into the complex plane. In the direct scattering problem, we study the analyticity, symmetries, and asymptotic behavior of Jost functions and the scattering matrix in detail. In addition, we find the discrete spectrum, residue conditions, trace formulas, and theta conditions in two cases: with simple poles and with second-order poles present in the spectrum. We solve the inverse problems using the RH problem formulated in terms of Jost functions and scattering coefficients. For further studying the structure of the soliton waves, we consider the dynamical behavior of soliton solutions for the mLL equation with reflectionless potentials. We graphically analyze some remarkable characteristics of these soliton solutions. Based on the analytic solutions, we discuss the influence of each parameter on the dynamics of the soliton waves and breather waves and propose a method for controlling such nonlinear phenomena.
Fundamental Research Funds for the Central Universities of China
2019ZDPY07 2019QNA35
Assistance Program for Future Outstanding Talents of China University of Mining and Technology
2020WLJCRCZL031
Postgraduate Research and Practice Innovation Program of Jiangsu Province
KYCX20_2038
This research was supported by the National Natural
Science Foundation of China (Grant No. 11975306), the Natural Science
Foundation of Jiangsu Province (Grant No. BK20181351), the Six Talent Peaks
Project in Jiangsu Province (Grant No. JY-059), the Fundamental Research
Fund for the Central Universities (Grant Nos. 2019ZDPY07 and 2019QNA35), the Assistance Program for Future Outstanding Talents of China University of
Mining and Technology (Grant No. 2020WLJCRCZL031), and the Postgraduate
Research & Practice Innovation Program of Jiangsu Province (Grant
No. KYCX20_2038).
\Bibitem{YanTia20}
\by Jin-Jie~Yang, Shou-Fu~Tian
\paper Riemann-Hilbert problem for the modified Landau-Lifshitz equation with nonzero boundary conditions
\jour TMF
\yr 2020
\vol 205
\issue 3
\pages 420--450
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\jour Theoret. and Math. Phys.
\yr 2020
\vol 205
\issue 3
\pages 1611--1637
\crossref{https://doi.org/10.1134/S0040577920120053}
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This publication is cited in the following 7 articles:
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Yilin Wang, Biao Li, “Riemann–Hilbert approach and soliton solutions for the Lakshmanan–Porsezian–Daniel equation with nonzero boundary conditions”, Commun. Theor. Phys., 76:11 (2024), 115003
Y. Huang, J. Di, Y. Yao, “The ${\bar{\partial }}$-dressing method applied to nonlinear defocusing Hirota equation with nonzero boundary conditions”, Nonlinear Dyn., 111:4 (2023), 3689
X.-F. Zhang, S.-F. Tian, “Riemann–Hilbert problem for the Fokas–Lenells equation in the presence of high-order discrete spectrum with non-vanishing boundary conditions”, Journal of Mathematical Physics, 64:5 (2023), 051503
J.-J. Yang, Sh.-F. Tian, Zh.-Q. Li, “Riemann-Hilbert problem for the focusing nonlinear Schrodinger equation with multiple high-order poles under nonzero boundary conditions”, Physica D, 432 (2022), 133162
J. Li, T. Xia, “The N-soliton solutions to the M-components nonlinear Schrödinger equations by the Riemann–Hilbert approach”, Partial Differential Equations in Applied Mathematics, 5 (2022), 100260
Jia Cheng, Shou-Fu Tian, Zhi-Jia Wu, “On the $\bar\partial$-problem and dressing method for the complex vector modified KdV equation”, Theoret. and Math. Phys., 209:2 (2021), 1579–1598
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