A Note on “New Hierarchies of Integrable Lattice Equations and Associated Properties: Darboux Transformation Conservation Laws and Integrable Coupling” [Rep. Math. Phys. 67 (2011), 259]

2016 ◽  
Vol 78 (3) ◽  
pp. 339-343
Author(s):  
Xi-Xiang Xu
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Integrable Lattice ◽  
Integrable Coupling ◽  
Math Phys

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].

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.

An integrable lattice hierarchy based on Suris system: $${\varvec{N}}$$ N -fold Darboux transformation and conservation laws

2017 ◽  
Vol 91 (1) ◽  
pp. 625-639 ◽  
Author(s):  
Qian Li ◽  
Deng-Shan Wang ◽  
Xiao-Yong Wen ◽  
Jian-Hong Zhuang
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Integrable Lattice

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.

scholarly journals Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Qianqian Yang ◽  
Qiulan Zhao ◽  
Xinyue Li
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Explicit Solutions ◽  
Liouville Integrability ◽  
Lattice Equation ◽  
Integrable Lattice ◽  
Matrix Spectral Problem

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

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.

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