An integrable lattice hierarchy associated with a 4 × 4 matrix spectral problem: N-fold Darboux transformation and dynamical properties

2020 ◽  
Vol 387 ◽  
pp. 124525 ◽  
Author(s):  
Ling Liu ◽  
Xiao-Yong Wen ◽  
Nan Liu ◽  
Tao Jiang ◽  
Jin-Yun Yuan
Keyword(s):  
Spectral Problem ◽  
Darboux Transformation ◽  
Integrable Lattice ◽  
Dynamical Properties ◽  
Matrix Spectral Problem

Analytic solutions and Darboux transformation to a new Hamiltonian lattice hierarchy

2016 ◽  
Vol 30 (08) ◽  
pp. 1650100 ◽  
Author(s):  
Shou-Fu Tian ◽  
Fu-Bao Zhou ◽  
Sheng-Wu Zhou ◽  
Tian-Tian Zhang
Keyword(s):  
Spectral Problem ◽  
Darboux Transformation ◽  
Analytical Solutions ◽  
Auxiliary Problem ◽  
Analytic Solutions ◽  
Graphical Analysis ◽  
Hamiltonian Lattice ◽  
Integrable Lattice ◽  
Matrix Spectral Problem

In this paper, a new Hamiltonian lattice hierarchy is analytically investigated, which can be reduced to some classic integrable lattice hierarchies, such as Ablowitz–Ladik hierarchy, Volterra hierarchy and multi-Hamiltonian lattice hierarchy, etc. By choosing the auxiliary problem [Formula: see text], we present a Darboux transformation (DT) to the new discrete matrix spectral problem. As its applications, a series of analytical solutions are generated in a recursive manner. Finally, the graphical analysis of these analytical solutions are presented, respectively. The DT of other lattice hierarchies can be also constructed in this method.

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.

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.

scholarly journals Conservation Laws and Self-Consistent Sources for an Integrable Lattice Hierarchy Associated with a Three-by-Three Discrete Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yu-Qing Li ◽  
Bao-Shu Yin
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Integrable Lattice ◽  
Self Consistent ◽  
Matrix Spectral Problem

A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

A HIERARCHY OF LIOUVILLE INTEGRABLE LATTICE EQUATION ASSOCIATED WITH A THREE-BY-THREE DISCRETE SPECTRAL PROBLEM AND ITS INFINITELY MANY CONSERVATION LAWS

2011 ◽  
Vol 25 (18) ◽  
pp. 2481-2492
Author(s):  
YU-QING LI ◽  
XI-XIANG XU
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Trace Identity ◽  
Lattice Equation ◽  
Hamiltonian Structures ◽  
Discrete Soliton ◽  
Soliton Equations ◽  
Integrable Lattice ◽  
Matrix Spectral Problem ◽  
Liouville Integrable

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

DARBOUX TRANSFORMATION OF THE MODIFIED TODA LATTICE EQUATION

2006 ◽  
Vol 20 (11) ◽  
pp. 641-648 ◽  
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN
Keyword(s):  
Spectral Problem ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Lattice Equation ◽  
Matrix Spectral Problem ◽  
Toda Lattice Equation

A modified Toda lattice equation associated with a properly discrete matrix spectral problem is introduced. Darboux transformation for the resulting lattice equation is constructed. As an application, the soliton solution for the Toda lattice equation is explicitly given.

THE NEGATIVE AND POSITIVE INTEGRABLE COUPLING HIERARCHIES DERIVED FROM A FOUR BY FOUR MATRIX SPECTRAL PROBLEM

2010 ◽  
Vol 24 (30) ◽  
pp. 2955-2970
Author(s):  
XI-XIANG XU
Keyword(s):  
Hamiltonian System ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Hamiltonian Form ◽  
Liouville Integrability ◽  
Positive Power ◽  
Integrable Lattice ◽  
Lax Operator ◽  
Matrix Spectral Problem ◽  
Integrable Coupling

Discrete integrable coupling hierarchies of two existing integrable lattice families are derived from a four by four discrete matrix spectral problem. It is shown that the obtained integrable coupling hierarchies respectively corresponds to negative and positive power expansions of the Lax operator with respect to the spectral parameter. Then, the Hamiltonian form of the negative integrable coupling hierarchy is constructed by using the discrete variational identity. Finally, Liouville integrability of each obtained discrete Hamiltonian system is demonstrated.

A FAMILY OF DISCRETE HAMILTONIAN EQUATIONS ASSOCIATED WITH A DISCRETE THREE-BY-THREE MATRIX SPECTRAL PROBLEM

2009 ◽  
Vol 23 (19) ◽  
pp. 3859-3869
Author(s):  
LIN-LIN MA ◽  
XI-XIANG XU
Keyword(s):  
Spectral Problem ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Integrable Lattice ◽  
The Family ◽  
Matrix Spectral Problem

A family of integrable lattice equations with four potentials is constructed from a new discrete three-by-three matrix spectral problem. The Hamiltonian structures of the integrable lattice equations in the family are derived by applying the discrete trace identity. Finally, infinitely many common commuting conserved functionals of the resulting integrable lattice equations are given.

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