Darboux transformation and exact solution to the nonlocal Kundu-Eckhaus equation

Article Mathematics, Applied

Darboux transformation to the nonlocal complex short pulse equation

Jian Li et al.

Summary: In this paper, a nonlocal complex short pulse equation is proposed and the Darboux transformation is constructed for it. By utilizing a seed solution, solutions composed of periodic waves and rogue waves are obtained.

APPLIED MATHEMATICS LETTERS (2022)

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Article Physics, Mathematical

Nonlocal symmetries and Darboux transformations of the Camassa-Holm equation and modified Camassa-Holm equation revisited

Nianhua Li et al.

Summary: This article introduces a method to incorporate wave functions into Darboux transformations and connects the nonlocal symmetries of the Camassa-Holm equation and the modified Camassa-Holm equation with those of the negative flows in the KdV hierarchy.

JOURNAL OF MATHEMATICAL PHYSICS (2022)

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Article Mathematics, Applied

Multi-component generalized Gerdjikov-Ivanov integrable hierarchy and its Riemann-Hilbert problem

Tongshuai Liu et al.

Summary: In this paper, the multi-component generalized Gerdjikov-Ivanov integrable hierarchy is obtained using the spectral problem and zero curvature equation. The multi-Hamiltonian structures of this hierarchy are investigated using trace identity. A specific Riemann-Hilbert problem is formulated for the generalized Gerdjikov-Ivanov integrable hierarchy, and by solving this problem, N-soliton solutions can be derived.

NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS (2022)

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Article Optics

The N-fold degenerate Darboux transformation of the three component Kundu-Eckhaus equations: Novel interactions between soliton and solution

Fengjiao Wu et al.

Summary: This paper derived an explicit representation of the N-fold degenerate Darboux transformation for the three component Kundu-Eckhaus equations, which model the effect of quintic nonlinearity on ultra-short optical pulse propagation in non-Kerr media. The DT is expressed by the initial eigenfunctions, spectral parameters and 'seed' solution, and the interactions between soliton and positon are discussed in detail, including elastic interactions, inelastic interactions, and bound states.

OPTIK (2021)

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Article Mathematics, Applied

Riemann-Hilbert approach to the modified nonlinear Schrodinger equation with non-vanishing asymptotic boundary conditions

Yiling Yang et al.

Summary: The paper presents the inverse scattering transform for the modified nonlinear Schrodinger equation and obtains N-soliton solutions through a Riemann-Hilbert problem. The dynamic features of the solutions and the impact of spectrum distribution and non-vanishing boundary conditions on soliton solutions are analyzed. Furthermore, differences between the results with non-vanishing boundaries and those on zero boundary conditions are discussed.

PHYSICA D-NONLINEAR PHENOMENA (2021)

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Article Materials Science, Multidisciplinary

Darboux transformation and exact solutions for a four-component Fokas-Lenells equation

Yihao Li et al.

Summary: This paper introduces a 4 x 4 matrix spectral problem and derives a four-component Fokas-Lenells equation, for which N-fold Darboux transformations are constructed. By utilizing limit technique and Taylor series, the N-fold generalized Darboux transformation is obtained, allowing for exact solutions of the four-component Fokas-Lenells equation, including soliton, breather, and rogue wave solutions, to be found. The dynamical behaviour of these solutions is then analyzed in detail.

RESULTS IN PHYSICS (2021)

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Article Engineering, Mechanical

Rogue wave and multi-pole solutions for the focusing Kundu-Eckhaus Equation with nonzero background via Riemann-Hilbert problem method

Ning Guo et al.

Summary: In this study, the inverse scattering transform method is used to solve the inverse problem of the focusing Kundu-Eckhaus equation with nonzero background at infinity, obtaining general multi-pole solutions and formulas for the N simple-pole soliton solutions. Peregrine's rational solution is shown to be an appropriate limit of the simple-pole soliton solutions at branch points, and taking proper limits can yield double- and triple-pole soliton solutions for the equation. The effect of the parameter beta and typical collisions between solutions are also graphically displayed.

NONLINEAR DYNAMICS (2021)

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Article Mathematics, Applied