scholarly journals A New Discrete Integrable System Derived from a Generalized Ablowitz-Ladik Hierarchy and Its Darboux Transformation

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Exact Solutions ◽  
Integrable System ◽  
Darboux Transformation ◽  
Dynamic Characteristics ◽  
Discrete System ◽  
Soliton Solutions ◽  
Interesting Phenomenon ◽  
Discrete Integrable System ◽  
New System

We find an interesting phenomenon that the discrete system appearing in a reference can be reduced to the old integrable system given by Merola, Ragnisco, and Tu in another reference. Differing from the works appearing in the above two references, a new discrete integrable system is obtained by the generalized Ablowitz-Ladik hierarchy; the Darboux transformation of this new discrete integrable system is established further. As applications of this Darboux transformation, different kinds of exact solutions of this new system are explicitly given. Investigatingthe properties of these exact solutions, we find that these exact solutions are not pure soliton solutions, but their dynamic characteristics are very interesting.

scholarly journals A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2315
Author(s):  
Meng-Li Qin ◽  
Xiao-Yong Wen ◽  
Manwai Yuen
Keyword(s):  
Exact Solutions ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Lattice System ◽  
Arbitrary Parameter ◽  
Rational Solutions ◽  
Limit States ◽  
Hybrid Solutions ◽  
Relativistic Toda Lattice

This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

Darboux Transformation and Symbolic Computation on Multi-Soliton and Periodic Solutions for Multi-Component Nonlinear Schr¨odinger Equations in an Isotropic Medium

2009 ◽  
Vol 64 (5-6) ◽  
pp. 300-308 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Tao Xu ◽  
Juan Li ◽  
Li-Li Li ◽  
Cheng Zhang ◽  
...  
Keyword(s):  
Periodic Solutions ◽  
Iterative Algorithm ◽  
Integrable System ◽  
Symbolic Computation ◽  
Darboux Transformation ◽  
Isotropic Medium ◽  
Soliton Solutions ◽  
Practical Interest ◽  
Lax Pair ◽  
Block Matrices

Abstract The Darboux transformation is applied to a multi-component nonlinear Schr¨odinger system, which governs the propagation of polarized optical waves in an isotropic medium. Based on the Lax pair associated with this integrable system, the formula for the n-times iterative Darboux transformation is constructed in the form of block matrices. The purely algebraic iterative algorithm is carried out via symbolic computation, and two different kinds of solutions of practical interest, i. e., bright multi-soliton solutions and periodic solutions, are also presented according to the zero and nonzero backgrounds.

Darboux transformation and exact solutions for a (2 + 1)-dimensional integrable system of nonlinear evolution equations

2020 ◽  
Vol 34 (34) ◽  
pp. 2050392
Author(s):  
Zhen Chuan Zhou ◽  
Xiao Ming Zhu
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Integrable System ◽  
Darboux Transformation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lax Pair ◽  
Nonlinear Evolution Equations ◽  
Recursion Operator

In this paper, starting from a spectral problem, we construct a [Formula: see text]-dimensional integrable system of nonlinear evolution equations. Based on the Lax pair, the recursion operator and Darboux transformation for the whole hierarchy were constructed. As an application, some exact solutions for the hierarchy are obtained by using the Darboux transformation.

Darboux transformation and the exact solutions of the (2+1)-dimensional nonlocal complex modified Korteweg-de Vries and Maxwell–Bloch equations

2020 ◽  
Vol 34 (22) ◽  
pp. 2050230
Author(s):  
Na-Na Li ◽  
Hui-Qin Hao ◽  
Rui Guo
Keyword(s):  
Exact Solutions ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Darboux Transformations ◽  
Bloch Equations ◽  
Dynamic Behaviors ◽  
De Vries ◽  
Maxwell Bloch Equations

In this paper, we consider the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlocal complex modified Korteweg-de Vries and Maxwell–Bloch (cmKdV-MB) equations. According to the relevant Lax pair presented, we construct one- and two-fold Darboux transformations (DT). The exact solutions are derived from the trivial seeds by DT and the dynamic behaviors of soliton solutions are analyzed by individual pictures.

Exact solutions of the noncommutative and commutative nonlinear Schrödinger equation in 2 + 1 dimensions

2017 ◽  
Vol 32 (29) ◽  
pp. 1750158 ◽  
Author(s):  
H. Sarfraz ◽  
U. Saleem
Keyword(s):  
Schrödinger Equation ◽  
Exact Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Localized Solutions ◽  
Multiple Soliton Solutions

In this paper, we presented a noncommutative (NC) generalization of nonlinear Schrödinger equation (NLSE) in 2 + 1 dimensions. A matrix Darboux transformation (MDT) is used to generate multiple soliton solutions for NC-NLSE and commutative NLSE in 2 + 1 dimensions. We expressed multiple soliton solutions in terms of quasideterminants and as ratios of ordinary determinants for NC and commutative NLSE in 2 + 1 dimensions, respectively. The quasideterminant formula for K-times repeated MDT enables us to compute single, double and triple soliton solutions for NC and commutative (2 + 1)-dimensional NLSE. Some interesting localized solutions are obtained for the NC and commutative NLSE in 2 + 1 dimensions.

scholarly journals Darboux Transformation, Exact Solutions and Conservation Laws for the Reverse Space-Time Fokas-Lenells Equation

2021 ◽  
Author(s):  
Jiang-Yan Song ◽  
Yu Xiao ◽  
Chi-Ping Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Space Time ◽  
Local Equation ◽  
Seed Solution ◽  
Dynamical Behaviors ◽  
Local Case ◽  
Physical Phenomena

Abstract In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas-Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics, and related fields.

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.

Some exact solutions and infinite conservation laws of an extended KdV integrable system

2020 ◽  
Vol 34 (26) ◽  
pp. 2050285 ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang ◽  
Jianqin Mei
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Integrable System ◽  
Soliton Solutions ◽  
Dark Soliton ◽  
Similarity Solutions ◽  
The Self ◽  
Analysis Method ◽  
Rational Solutions ◽  
Adjoint Operators

In this paper, we investigate some symmetries and Lie-group transformations of an integrable system by using the symmetry analysis method. It follows that the resulting similarity solutions are obtained by applying the characteristic equations of the symmetries. By applying the software Maple,we work out some exact solutions of the generalized KdV system, including the rational solutions, the periodic solutions, the dark soliton solutions, and so on. Finally, we make use of the self-adjoint operators to investigate the nonlinear self-adjointness and the conservation laws of the generalized KdV integrable system.

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

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