Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrödinger system in a birefringent optical fiber

Wang, Meng; Tian, Bo; Liu, Shaohua

doi:10.1016/j.aml.2020.106936

Cited by 100 publications

(12 citation statements)

References 29 publications

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“…In soliton theory, conservation laws also play an important role in integrability of soliton equations. [19,20] Conservation laws are a reaction to the phenomenon that certain physical amounts are not changed with time. The infinite conservation laws are intimately linked with the existence of solitary particles.…”

Section: Introductionmentioning

confidence: 99%

Darboux transformation, infinite conservation laws, and exact solutions for the nonlocal Hirota equation with variable coefficients

Liu 刘,

Yan 闫,

Jin 金

et al. 2023

Chinese Phys. B

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This article presents the construction of a nonlocal Hirota equation with variable coefficient and its Darboux transformation. Using zero-seed solutions, 1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation, along with the expression for N-soliton solutions. The influence of coefficient functions on the solutions is investigated by choosing different coefficient functions, and the dynamics of the solutions are analyzed. For the first time, this article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations. The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

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Section: Introductionmentioning

confidence: 99%

Darboux transformation, infinite conservation laws, and exact solutions for the nonlocal Hirota equation with variable coefficients

Liu 刘,

Yan 闫,

Jin 金

et al. 2023

Chinese Phys. B

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“…[16][17][18] Some recent achievements about the similarity reduction, Bäcklund transformation, Lax pair, Darboux transformation of the Schrödinger-type model, Hirota system, dispersive long-wave system, and generalized nonlinear evolution equation can be seen in previous studies. [19][20][21][22][23] However, the rogue wave formed in the ocean must be described by the (2 + 1)-dimensional or (3 + 1)-dimensional models regardless (1 + 1)-dimensional ones. 24 Recently, Shan has applied the scaling transformation to a generalized (2 + 1)-dimensional dispersive long-wave system to derive N-soliton solution with the help of Bell polynomials and symbolic computation.…”

Section: Introductionmentioning

confidence: 99%

“…Besides, the physics‐informed neural networks (PINNs) deep learning method has been applied to NLSE by Chen to derive rational solution, periodic solution, rogue wave solution, and rogue periodic wave solution (rogue wave on the periodic background) in complex space 16–18 . Some recent achievements about the similarity reduction, Bäcklund transformation, Lax pair, Darboux transformation of the Schrödinger‐type model, Hirota system, dispersive long‐wave system, and generalized nonlinear evolution equation can be seen in previous studies 19–23 . However, the rogue wave formed in the ocean must be described by the (2 + 1)‐dimensional or (3 + 1)‐dimensional models regardless (1 + 1)‐dimensional ones 24 .…”

Section: Introductionmentioning

confidence: 99%

Rational and semi‐rational solutions for a (3 + 1)‐dimensional generalized KP–Boussinesq equation in shallow water wave

Yan

Xie

2022

Math Methods in App Sciences

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In this paper, a new extended (3 + 1)-dimensional generalized Kadomtsev-Petviashvili (KP)-Boussinesq equation, with two additional terms u tt and u xz , is proposed and investigated. This new equation models both left and right going waves like the Boussinesq equation and describes the transmission of tsunami waves at the bottom of the ocean and nonlinear ion-acoustic waves in the magnetized dusty plasm. We have constructed one set of breathers and first order periodic waves from the two-soliton solution. Moreover, the four-soliton solution consists of two sets of breathers, second order periodic waves, and a hybrid of breathers and periodic line waves. Then, we introduce two kinds of "long wave" limits to obtain the rational and semi-rational solutions for N = 2, 3, 4. Under specific parametric constraints, the obtained rational and semi-rational solutions are nonsingular. The rational solution can be classified as first order line rogue wave, single breather, second order line rogue wave, double breather, a hybrid of breather and line rogue wave, a hybrid of breather, and single soliton. The semi-rational solution can be classified as the first and second order kink-shaped rogue wave, a hybrid of breather and one (two) soliton(s), a hybrid of a set of breathers, and a single soliton (breather). In addition, we give Theorem 2.1 for the higher order rational solutions.

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“…The central region of the b-p solution exhibits a rogue wave like structure. The nonlinear Schrödinger equation and its higher order generalizations have been considered in several fields including fluid dynamics, birefringent optical fiber, shallow water surfaces, Heisenberg spin systems and ocean waves [39][40][41][42][43][44][45][46]. In this work, we construct certain smooth positons and b-p solutions of an extended nonlinear Schrödinger equation (ENLSE) [47][48][49][50] with the cubic and quartic nonlinearity.…”

Section: Introductionmentioning

confidence: 99%

Higher order smooth positon and breather positon solutions of an extended nonlinear Schrödinger equation with the cubic and quartic nonlinearity

Monisha

Priya²,

Senthilvelan³

et al. 2022

Chaos, Solitons & Fractals

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Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrödinger system in a birefringent optical fiber

Cited by 100 publications

References 29 publications

Darboux transformation, infinite conservation laws, and exact solutions for the nonlocal Hirota equation with variable coefficients

Darboux transformation, infinite conservation laws, and exact solutions for the nonlocal Hirota equation with variable coefficients

Rational and semi‐rational solutions for a (3 + 1)‐dimensional generalized KP–Boussinesq equation in shallow water wave

Higher order smooth positon and breather positon solutions of an extended nonlinear Schrödinger equation with the cubic and quartic nonlinearity

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