Riemann–Hilbert approach for discrete sine‐Gordon equation with simple and double poles

Chen, Meisen; Fan, Engui

doi:10.1111/sapm.12472

Cited by 22 publications

(4 citation statements)

References 36 publications

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“…The explict N-triple-pole solutions have been derived in terms of the determinants for the focusing NLS hierarchy with NZBCs [52] and the multiple high-order poles solutions have been obtained for the focusing NLS equation in virtue of the Laurent expansion and Taylor series to avoid consideration of the residue conditions [53]. The derivative NLS equation [54], the discrete Sine-Gordon equation [55], the Kundu-Eckhaus equation [56] with NZBCs et al also have been studied via the RH approach. The objective of our work is not only to study the ISTs with NZBCs for the nonlocal LPD equation via RH approach, but also to obtain abundant multi-pole solitons and breathers and reveal their properties.…”

Section: Introductionmentioning

confidence: 99%

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Qin,

2024

Phys. Scr.

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Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

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Section: Introductionmentioning

confidence: 99%

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Qin,

2024

Phys. Scr.

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“…The most important of these studies are meant to obtain soliton solutions. Many methods have been developed, including the inverse scattering transformation [7,8], the Darboux transformation [9], Bäcklund transformations [10], and Hirota's bilinear technique [11].…”

Section: Introductionmentioning

confidence: 99%

Lump-Type Solutions, Mixed Solutions and Rogue Waves for a (3+1)-Dimensional Variable-Coefficients Burgers Equation

Wu,

Cai,

Cheng

2024

Symmetry

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In this work, we consider the (3+1)-dimensional Burgers equation with variable coefficients, which is frequently used to define the motion of solitary waves. Abundant lump waves are constructed by taking the ansatz as a rational function. Furthermore, mixed solutions utilizing lump waves, rogue waves, and kink solitons are obtained by combining the rational function with an exponential function, resulting in fission and fusion phenomena.

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“…[19] Correspondingly, its discrete forms may also well describe certain physical phenomena. Some methods for obtaining the exact solutions of discrete integrable equations include the inverse scattering method, [20,21] Riemann-Hilbert method, [22,23] algebraic geometry method, [24] Darboux transformation (DT), [16,25,26] and so on. Among them, the DT is one of the available methods, whose basic idea is to keep the Lax pair of discrete equations covariant.…”

Section: Introductionmentioning

confidence: 99%

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Liu 刘,

Wen 闻

2023

Chinese Phys. B

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Under consideration in this paper is the discrete coupled mKdV equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N−r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem for the first time, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results might be helpful for understanding some physical phenomena in the field of shallow water wave, optics and plasma.

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Riemann–Hilbert approach for discrete sine‐Gordon equation with simple and double poles

Cited by 22 publications

References 36 publications

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Lump-Type Solutions, Mixed Solutions and Rogue Waves for a (3+1)-Dimensional Variable-Coefficients Burgers Equation

Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

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