scholarly journals Lax pair, conservation laws and Darboux transformation of the high-order Lax equation in fluid dynamics

2017 ◽  
Author(s):  
Wenxin Zheng ◽  
Guangmei Wei
Keyword(s):  
Fluid Dynamics ◽  
Conservation Laws ◽  
Darboux Transformation ◽  
Lax Pair ◽  
High Order ◽  
Lax Equation

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.

N-fold Darboux transformation and exact solutions of the Suris system

2018 ◽  
Vol 32 (09) ◽  
pp. 1850019 ◽  
Author(s):  
Qian Li ◽  
Minghui Liu ◽  
Deng-Shan Wang ◽  
Xiao-Yong Wen
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Lax Pair ◽  
Nonlinear Structures ◽  
Physical Phenomena

In this paper, the N-fold Darboux transformation of the Suris system is established by gauge transformation of the Lax pair. As a result, the N-fold exact solutions of the Suris system are derived in terms of the determinant. It is shown that this system can support certain abundant and peculiar nonlinear structures, which may explain some interesting physical phenomena. Moreover, the infinitely many conservation laws of the Suris system are given.

scholarly journals Darboux transformation, bright and dark-bright solitons of an N-coupled high-order nonlinear Schrödinger system in an optical fiber

2021 ◽  
Author(s):  
Xi-Hu Wu ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Cui-Cui Ding ◽  
Fei-Yan Liu ◽  
...  
Keyword(s):  
Optical Fiber ◽  
Asymptotic Analysis ◽  
Darboux Transformation ◽  
Lax Pair ◽  
Second Order ◽  
High Order ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Bright Solitons ◽  
Schrödinger System

Abstract In this paper, an N -coupled high-order nonlinear Schrödinger system, which describes the properties of the ultrashort optical pulses in an optical fiber, is investigated with the help of Darboux transformation (DT) method and asymptotic analysis. Starting from the given (2 N +1)th-order Lax pair, we construct a new form of DT (complex eigenfunctions of Lax pair involved) to derive the formulas of the n th-iterated solutions, where n and N are the positive integers. On the zero background, the first- and second-order solitons are obtained and analysed through the asymptotic analysis. Multi-parameter adjustment is proceeded since there are 3 N +4 real parameters in the second-order solitons. We find that under certain conditions each of the two interaction patterns (elastic, inelastic) holds in the second-order soliton. On the plane wave background, the first-order bright and dark-bright solitons are obtained. Soliton velocities, amplitudes, widths and characteristic lines of the first-order bright and dark-bright solitons are presented and analysed.

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.

Conservation laws and Darboux transformations for a 3-coupled integrable lattice equations

2020 ◽  
Vol 34 (21) ◽  
pp. 2050218
Author(s):  
Fangcheng Fan ◽  
Shaoyun Shi ◽  
Zhiguo Xu
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Polynomial Form ◽  
Darboux Transformations ◽  
Logarithmic Form ◽  
Integrable Lattice

In this paper, we firstly establish infinitely many conservation laws of the 3-coupled integrable lattice equations by using the Riccati method. Comparing with the results obtained by Sahadevan and Balakrishnan, we not only get infinite conserved densities of the polynomial form, but also some conserved densities of logarithmic form. Secondly, Darboux transformation for the system is derived with the help of the Lax pair and gauge transformation. Finally, we obtain the exact solutions of the system with the obtained Darboux transformation, and present the soliton solutions and their figures with properly parameters.

High-order rogue-wave of the inhomogeneous nonlinear Hirota equation with a self-consistent source

2019 ◽  
Vol 33 (08) ◽  
pp. 1950087 ◽  
Author(s):  
Yuqin Yao ◽  
Yehui Huang
Keyword(s):  
Wave Propagation ◽  
Optical Fibers ◽  
Darboux Transformation ◽  
Rogue Wave ◽  
Lax Pair ◽  
High Order ◽  
Hirota Equation ◽  
Wave Solutions ◽  
Self Consistent ◽  
Evolution Dynamics

The inhomogeneous nonlinear Hirota equation is more realistic than the nonlinear Schrödinger equation when approximating the wave propagation in the ocean and optical fibers. In this paper, the inhomogeneous nonlinear Hirota equation with a self-consistent source and its Lax pair are presented. The high-order rogue-wave solutions are worked out by the generalized Darboux transformation and their evolution dynamics is analyzed.

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