Quasi-Grammian solutions of the generalized Heisenberg magnet model

2020 ◽  
Vol 98 (3) ◽  
pp. 303-311
Author(s):  
Z. Amjad ◽  
B. Haider
Keyword(s):  
Darboux Transformation ◽  
Lie Group ◽  
Soliton Solutions ◽  
Two Dimensions ◽  
Explicit Solutions ◽  
Heisenberg Magnet ◽  
Model Based ◽  
Binary Darboux Transformation

In this paper we use standard binary Darboux transformation to obtain the quasi-Grammian multi-soliton solutions of generalized Heisenberg magnet model in two dimensions. We also discuss the model based on the Lie group SU(n) and obtain explicit solutions of the model for the SU(2) case.

DARBOUX TRANSFORMATION AND MULTI-SOLITON SOLUTIONS OF PRINCIPAL CHIRAL AND WZW MODELS

2011 ◽  
Vol 26 (01) ◽  
pp. 73-85 ◽  
Author(s):  
U. SALEEM ◽  
M. HASSAN
Keyword(s):  
Darboux Transformation ◽  
Lie Group ◽  
Soliton Solutions ◽  
Two Dimensions ◽  
Model Based ◽  
Conserved Currents

In this paper we present Darboux transformation for the principal chiral and WZW models in two dimensions and construct multi-soliton solutions in terms of quasideterminants. We also establish the Darboux transformation on the holomorphic conserved currents of the WZW model and expressed them in terms of the quasideterminant. We discuss the model based on the Lie group SU (n) and obtain explicit soliton solutions for the SU(2) model.

ON THE DRESSING METHOD FOR THE GENERALIZED COUPLED DISPERSIONLESS INTEGRABLE SYSTEM

2013 ◽  
Vol 28 (20) ◽  
pp. 1350088 ◽  
Author(s):  
NOSHEEN MUSHAHID ◽  
MAHMOOD UL HASSAN
Keyword(s):  
Integrable System ◽  
Lie Group ◽  
Matrix Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Two Dimensions ◽  
Nonlinear Matrix Equation ◽  
Dressing Method ◽  
Nonlinear Matrix ◽  
Funct Anal

The dressing method of Zakharov and Shabat [Funct. Anal. Appl.8, 226 (1974) and ibid.13, 166 (1980)] has been employed to the generalized coupled dispersionless integrable system in two dimensions. The dressed solutions to the Lax pair and to the nonlinear matrix equation have been obtained in terms of Hermitian projectors. The dressing method has been related with the quasi-determinant solutions obtained by using the standard matrix Darboux transformation. The iteration of dressing procedure has been shown to give N-soliton solutions of the system. At the end, the explicit soliton solution has been obtained for the system based on Lie group SU(2).

scholarly journals Darboux Transformation and Explicit Solutions for a Generalized Sawada-Kotera Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Guo-Liang He ◽  
Ting Su
Keyword(s):  
Periodic Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Explicit Solutions ◽  
Rational Solutions ◽  
Lax Pairs ◽  
Spectral Problems

A generalized Sawada-Kotera equation and its Lax pairs are proposed. With the help of the gauge transformation between spectral problems, a Darboux transformation for the generalized SK equation is constructed. As an application of the Darboux transformation, we give some explicit solutions of the generalized SK equation such as the rational solutions, soliton solutions, and periodic solutions.

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.

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 Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Y Hanif ◽  
U Saleem
Keyword(s):  
Darboux Transformation ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Dynamical Features ◽  
Discrete Darboux Transformation ◽  
Explicit Expressions

Abstract We study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

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