Discrete N-fold Darboux transformation and infinite number of conservation laws of a four-component Toda lattice

2021 ◽  
Author(s):  
Fangcheng Fan
Keyword(s):  
Solitary Wave ◽  
Conservation Laws ◽  
Darboux Transformation ◽  
Infinite Number ◽  
Iterative Process ◽  
Toda Lattice ◽  
Solitary Wave Solution ◽  
Wave Structure ◽  
Lax Pair ◽  
Wave Change

In this paper, we investigate a four-component Toda lattice (TL), which may be used to model the wave propagation in lattices just like the famous TL. By means of the Lax pair and gauge transformation, we construct the [Formula: see text]-fold Darboux transformation (DT), which enables us to obtain multi-soliton or multi-solitary wave solution without complex iterative process. Through the obtained DT, [Formula: see text]-fold explicit exact solutions of the system and their figures with proper parameters are presented from which we find the [Formula: see text]-fold solution shows two-solitary wave structure, the amplitude and shape of the wave change with time. Finally, we derive an infinite number of conservation laws formulaically to illustrate the integrability of the system.

Notes on “infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system [Int. J. Mod. Phys. B 33 (2019) 1950147]

2021 ◽  
pp. 2150141
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Infinite Number ◽  
Lattice System ◽  
Darboux Transformations ◽  
Integrable Lattice

We show that the Darboux transformation in “Infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system” [Int. J. Mod. Phys. B 33 (2019) 1950147] is incorrect, and construct a correct Darboux transformation.

N-fold Darboux transformation of a six-field integrable lattice system

2021 ◽  
pp. 2150248
Author(s):  
Yanan Qin
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Lattice System ◽  
Local Conservation ◽  
Integrable Lattice ◽  
Coupled Equation ◽  
Math Phys

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct [Formula: see text]-fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when [Formula: see text].

Lax Pair, Conservation Laws, Solitons, and Rogue Waves for a Generalised Nonlinear Schrödinger–Maxwell–Bloch System under the Nonlinear Tunneling Effect for an Inhomogeneous Erbium-Doped Silica Fibre

2016 ◽  
Vol 71 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Zhe Gao ◽  
Yi-Tian Gao ◽  
Chuan-Qi Su ◽  
Qi-Min Wang ◽  
Bing-Qing Mao
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Optical Pulse ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Pulse Frequency ◽  
Lax Pair ◽  
Tunneling Effect ◽  
Nonlinear Schrödinger ◽  
Erbium Doped

AbstractUnder investigation in this article is a generalised nonlinear Schrödinger-Maxwell-Bloch system for the picosecond optical pulse propagation in an inhomogeneous erbium-doped silica optical fibre. Lax pair, conservation laws, Darboux transformation, and generalised Darboux transformation for the system are constructed; with the one- and two-soliton solutions, the first- and second-order rogue waves given. Soliton propagation is discussed. Nonlinear tunneling effect on the solitons and rogue waves are investigated. We find that (i) the detuning of the atomic transition frequency from the optical pulse frequency affects the velocity of the pulse when the detuning is small, (ii) nonlinear tunneling effect does not affect the energy redistribution of the soliton interaction, (iii) dispersion barrier/well has an effect on the soliton velocity, whereas nonlinear well/barrier does not, (iv) nonlinear well/barrier could amplify/compress the solitons or rogue waves in a smoother manner than the dispersion barrier/well, and (v) dispersion barrier could “attract” the nearby rogue waves, whereas the dispersion well has a repulsive effect on them.

scholarly journals Infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system

2019 ◽  
Vol 33 (14) ◽  
pp. 1950147 ◽  
Author(s):  
Fangcheng Fan ◽  
Shaoyun Shi ◽  
Zhiguo Xu
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Lattice System ◽  
Darboux Transformations ◽  
Lattice Equation ◽  
Seed Solution ◽  
Integrable Lattice ◽  
Special Cases

In this paper, we study a 6-field integrable lattice system, which, in some special cases, can be reduced to the self-dual network equation, the discrete second-order nonlinear Schrödinger equation and the relativistic Volterra lattice equation. With the help of the Lax pair, we construct infinitely many conservation laws and a new Darboux transformation for system. Exact solutions resulting from the obtained Darboux transformation are presented by using a given seed solution. Further, we generate the soliton solutions and plot the figures of one-soliton solutions with properly parameters.

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