DARBOUX TRANSFORMATION AND EXPLICIT SOLUTIONS FOR A 3-FIELD INTEGRABLE LATTICE SYSTEM WITH THREE ARBITRARY CONSTANTS

2011 ◽  
Vol 25 (19) ◽  
pp. 2609-2619 ◽  
Author(s):  
XI-XIANG XU
Keyword(s):  
Darboux Transformation ◽  
Lax Pair ◽  
Lattice System ◽  
Explicit Solutions ◽  
Integrable Lattice ◽  
In Virtue Of

A 3-field integrable lattice system with three arbitrary constants and its Lax pair are presented. In virtue of the Lax pair, a Darboux transformation for the 3-field integrable lattice system is obtained, from which the explicit solutions of the 3-field integrable lattice system are given.

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].

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.

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.

scholarly journals Obtaining Solutions of a Vakhnenko Lattice System by N-Fold Darboux Transformation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Exact Solution ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Lax Pair ◽  
Transformation Matrix ◽  
Lattice System

Using a suitable gauge transformation matrix, we present a N -fold Darboux transformation for a Vakhnenko lattice system. This transformation preserves the form of Lax pair of the Vakhnenko lattice system. Applying the obtained Darboux transformation, we arrive at an exact solution of the Vakhnenko lattice system.

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.

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

2021 ◽  
Author(s):  
Zhiguo Xu
Keyword(s):  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Integrable Hierarchy ◽  
Integrable Lattice ◽  
Pass Through ◽  
Matrix Spectral Problem ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Hierarchy

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

Integrable variable-coefficient derivative Spin-1 Gross–Pitaevskii equations and their explicit solutions

2021 ◽  
Author(s):  
Ting Su ◽  
Junhong Yao ◽  
Yanan Huang
Keyword(s):  
Variable Coefficient ◽  
Separation Of Variables ◽  
Lax Pair ◽  
Explicit Solutions ◽  
Dressing Method

Based on the generalized dressing method, we propose a new integrable variable coefficient Spin-1 Gross–Pitaevskii equations and derive their Lax pair. Using separation of variables, we have derived explicit solutions of the equations. In order to analyze the characteristic of derived solution, the graphical wave of the solutions is plotted with the aid of Matlab.

Lax pair, Darboux transformation and rogue-periodic waves of a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics

2021 ◽  
pp. 2150451 ◽  
Author(s):  
Cheng-Cheng Wei ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Su-Su Chen ◽  
Dan-Yu Yang
Keyword(s):  
Darboux Transformation ◽  
Periodic Wave ◽  
Lax Pair ◽  
Third Order ◽  
Periodic Wave Solutions ◽  
The Third ◽  
Third Order Dispersion ◽  
Wave Solutions ◽  
Minimum Amplitude ◽  
Order Dispersion

For a nonlinear Schrödinger–Hirota equation with the spatio-temporal dispersion and Kerr law nonlinearity in nonlinear optics, we derive a Lax pair, a Darboux transformation and two families of the periodic-wave solutions via the Jacobian elliptic functions dn and cn. We construct the linearly-independent and non-periodic solutions of that Lax pair, and substitute those solutions into the Darboux transformation to get the rogue-periodic-wave solutions. When the third-order dispersion or group velocity dispersion (GVD) or inter-modal dispersion (IMD) increases, the maximum amplitude of the rogue-periodic wave remains unchanged. From the rogue-dn-periodic-wave solutions, when the GVD decreases, the minimum amplitude of the rogue-dn-periodic wave decreases. When the third-order dispersion decreases, the minimum amplitude of the rogue-dn-periodic wave rises. Decrease of the IMD causes the period of the rogue-dn-periodic wave to decrease. From the rogue-cn-periodic-wave solutions, when the GVD increases, the minimum amplitude of the rogue-cn-periodic wave decreases. Increase of the third-order dispersion or IMD leads to the decrease of the period.

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