THE GENERALIZED MULTI-COMPONENT AKNS HIERARCHY AND N-FOLD DARBOUX TRANSFORMATION

2006 ◽  
Vol 20 (25) ◽  
pp. 1575-1589 ◽  
Author(s):  
HONG-XIANG YANG ◽  
DAO-LIN WANG ◽  
CHANG-SHENG LI
Keyword(s):  
Spectral Problem ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Lax Pairs ◽  
Akns Hierarchy ◽  
Soliton Equations ◽  
Gauge Transformations ◽  
Set Up ◽  
Liouville Integrable

Starting from a 3×3 spectral problem, by using the Tu scheme, a hierarchy of generalized multi-component AKNS soliton equations are derived. It is shown that each equation in the resulting hierarchy is Liouville integrable. With the help of gauge transformations of the Lax pairs, an N-fold Darboux transformation (DT) with multi-parameters for the spectral problem is set up. For application, the soliton solutions of the first nonlinear soliton equation are explicitly given.

scholarly journals A Five-Component Generalized mKdV Equation and Its Exact Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1145
Author(s):  
Bo Xue ◽  
Huiling Du ◽  
Ruomeng Li
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Rational Functions ◽  
Mkdv Equation ◽  
Darboux Transformations ◽  
Lax Pairs ◽  
Gauge Transformations

In this paper, a 3 × 3 spectral problem is proposed and a five-component equation that consists of two different mKdV equations is derived. A Darboux transformation of the five-component equation is presented relating to the gauge transformations between the Lax pairs. As applications of the Darboux transformations, interesting exact solutions, including soliton-like solutions and a solution that consists of rational functions of e x and t, for the five-component equation are obtained.

Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions

2017 ◽  
Vol 72 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Lili Feng ◽  
Fajun Yu ◽  
Li Li
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Non Linear ◽  
And Mathematics

AbstractStarting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.

scholarly journals New Lax Pairs and Darboux Transformation and Its Application to a Shallow Water Wave Model of Generalized KdV Type

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Huizhang Yang ◽  
Weiguo Rui
Keyword(s):  
Shallow Water ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Water Wave ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Wave Model ◽  
Lax Pairs ◽  
Shallow Water Wave ◽  
Generalized Kdv Equation

New Lax pairs of a shallow water wave model of generalized KdV equation type are presented. According to this Lax pair, we constructed a new spectral problem. By using this spectral problem, we constructed Darboux transformation with the help of a gauge transformation. Applying this Darboux transformation, some new exact solutions including double-soliton solution of the shallow water wave model of generalized KdV equation type are obtained. In order to visually show dynamical behaviors of these double soliton solutions, we plot graphs of profiles of them and discuss their dynamical properties.

scholarly journals Darboux transformation for classical acoustic spectral problem

2003 ◽  
Vol 2003 (49) ◽  
pp. 3123-3142 ◽  
Author(s):  
A. A. Yurova ◽  
A. V. Yurov ◽  
M. Rudnev
Keyword(s):  
Spectral Problem ◽  
Inhomogeneous Medium ◽  
Darboux Transformation ◽  
Maxwell Equations ◽  
Soliton Solutions ◽  
Moutard Transformation ◽  
Acoustic Problem ◽  
Spatial Dimensions ◽  
Dielectric Permitivity ◽  
Harry Dym Equation

We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows steps similar to those for the Schrödinger operator. However, there is no one-to-one correspondence between the two problems. The technique developed enables one to construct new families of integrable potentials for the acoustic problem, in addition to those already known. The acoustic problem produces a nonlinear Harry Dym PDE. Using the technique, we reproduce a pair of simple soliton solutions of this equation. These solutions are further used to construct a new positon solution for this PDE. Furthermore, using the dressing-chain approach, we build a modified Harry Dym equation together with its LA pair. As an application, we construct some singular and nonsingular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.

Decomposition, Darboux transformation and soliton solutions for the (2 + 1)-dimensional integrable nonlocal “breaking soliton” equation

2020 ◽  
Vol 34 (24) ◽  
pp. 2050251
Author(s):  
Xiaoming Zhu ◽  
Kelei Tian
Keyword(s):  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Interaction Mechanisms ◽  
Dark Solitons ◽  
De Vries ◽  
Nonlocal Nonlinear Schrödinger Equation ◽  
Nonlocal Nonlinear

In this paper, we investigate an integrable nonlocal “breaking soliton” equation, which can be decomposed into the nonlocal nonlinear Schrödinger equation and the nonlocal complex modified Korteweg–de Vries equation. As an application, with the use of this decomposition and Darboux transformation, the dark solitons, antidark solitons, rational dark solitons and rational antidark solitons of the considered equation are given explicitly. In particular, the interaction mechanisms of these solutions are discussed and illustrated through some figures.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

2009 ◽  
Vol 23 (24) ◽  
pp. 4855-4879 ◽  
Author(s):  
HONWAH TAM ◽  
YUFENG ZHANG
Keyword(s):  
Lie Algebra ◽  
Spectral Radius ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Spectral Matrix ◽  
Akns Hierarchy ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Set Up

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

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.

A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations

2015 ◽  
Vol 16 (7-8) ◽  
pp. 301-306 ◽  
Author(s):  
Ning Zhang ◽  
Tiecheng Xia
Keyword(s):  
Eigenvalue Problem ◽  
Exact Solutions ◽  
Spectral Parameter ◽  
Darboux Transformation ◽  
Lattice Models ◽  
Darboux Transformations ◽  
Discrete Eigenvalue ◽  
Negative Power ◽  
Soliton Equations ◽  
Gauge Transformations

AbstractBy considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

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