Conservation Laws and $$\tau $$τ-Symmetry Algebra of the Gerdjikov–Ivanov Soliton Hierarchy

2018 ◽  
Vol 43 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Jian-bing Zhang ◽  
Yingyin Gongye ◽  
Wen-Xiu Ma
Keyword(s):  
Conservation Laws ◽  
Symmetry Algebra ◽  
Soliton Hierarchy

scholarly journals A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Yuqin Yao ◽  
Shoufeng Shen ◽  
Wen-Xiu Ma
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Matrix Spectral Problem

Associated withso~(3,R), a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.

scholarly journals Self-Consistent Sources and Conservation Laws for Nonlinear Integrable Couplings of the Li Soliton Hierarchy

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Han-yu Wei ◽  
Tie-cheng Xia
Keyword(s):  
Conservation Laws ◽  
Lie Algebras ◽  
Soliton Hierarchy ◽  
Self Consistent ◽  
Integrable Couplings ◽  
Integrable Coupling

New explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Li soliton hierarchy are obtained. Then, the nonlinear integrable couplings of Li soliton hierarchy with self-consistent sources are established. Finally, we present the infinitely many conservation laws for the nonlinear integrable coupling of Li soliton hierarchy.

scholarly journals A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 535 ◽  
Author(s):  
Jian-bing Zhang ◽  
Yingyin Gongye ◽  
Wen-Xiu Ma
Keyword(s):  
Vries Equation ◽  
Symmetry Algebra ◽  
Arbitrary Parameter ◽  
Generalized Derivative ◽  
Infinite Dimensional ◽  
Nonlinear Schrödinger ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
New Equation ◽  
Matrix Spectral Problem

A matrix spectral problem is researched with an arbitrary parameter. Through zero curvature equations, two hierarchies are constructed of isospectral and nonisospectral generalized derivative nonlinear schrödinger equations. The resulting hierarchies include the Kaup-Newell equation, the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation, the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation and a new equation as special reductions. The integro-differential operator related to the isospectral and nonisospectral hierarchies is shown to be not only a hereditary but also a strong symmetry of the whole isospectral hierarchy. For the isospectral hierarchy, the corresponding τ -symmetries are generated from the nonisospectral hierarchy and form an infinite-dimensional symmetry algebra with the K-symmetries.

scholarly journals Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. Wilson ◽  
A. H. Kara
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Wave Equations ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

scholarly journals A New Soliton Hierarchy Associated withso⁡3,Rand Its Conservation Laws

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Hanyu Wei ◽  
Tiecheng Xia ◽  
Guoliang He
Keyword(s):  
Conservation Laws ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Soliton Equations ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Orthogonal Lie Algebra

Based on the three-dimensional real special orthogonal Lie algebraso(3,R), we construct a new hierarchy of soliton equations by zero curvature equations and show that each equation in the resulting hierarchy has a bi-Hamiltonian structure and thus integrable in the Liouville sense. Furthermore, we present the infinitely many conservation laws for the new soliton hierarchy.

scholarly journals Lie Symmetry Analysis of C 1 m , a , b Partial Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Hengtai Wang ◽  
Aminu Ma’aruf Nass ◽  
Zhiwei Zou
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Symmetry Algebra ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Invariant Solutions ◽  
Lie Symmetry Analysis ◽  
Low Dimensions ◽  
Similarity Reductions

In this article, we discussed the Lie symmetry analysis of C 1 m , a , b fractional and integer order differential equations. The symmetry algebra of both differential equations is obtained and utilized to find the similarity reductions, invariant solutions, and conservation laws. In both cases, the symmetry algebra is of low dimensions.

scholarly journals Approximate Symmetry Analysis and Approximate Conservation Laws of Perturbed KdV Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Yu-Shan Bai ◽  
Qi Zhang
Keyword(s):  
Conservation Laws ◽  
Kdv Equation ◽  
Symmetry Algebra ◽  
Optimal System ◽  
Approximate Symmetry ◽  
Invariant Solutions ◽  
Approximate Symmetries ◽  
One Dimensional ◽  
Perturbed Kdv Equation ◽  
Partial Lagrangian

Approximate symmetries, which are admitted by the perturbed KdV equation, are obtained. The optimal system of one-dimensional subalgebra of symmetry algebra is obtained. The approximate invariants of the presented approximate symmetries and some new approximately invariant solutions to the equation are constructed. Moreover, the conservation laws have been constructed by using partial Lagrangian method.

An integrable soliton hierarchy associated with the Boiti–Pempinelli–Tu spectral problem

2021 ◽  
pp. 2150282
Author(s):  
Emmanuel A. Appiah ◽  
Solomon Manukure
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Lie Algebra ◽  
Mkdv Equation ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Soliton Equations ◽  
Soliton Hierarchy ◽  
Matrix Spectral Problem ◽  
Math Phys

Based on the Tu scheme [G.-Z. Tu, J. Math. Phys. 30 (1989) 330], we construct a counterpart of the Boiti–Pempinelli–Tu soliton hierarchy from a matrix spectral problem associated with the Lie algebra [Formula: see text], and formulate Hamiltonian structures for the resulting soliton equations by means of the trace identity. We then show that the newly presented equations possess infinitely many commuting symmetries and conservation laws. Finally, we derive the well-known combined KdV-mKdV equation from the new hierarchy.

Lie symmetry analysis and conservation laws for the time fractional Black–Scholes equation

2019 ◽  
Vol 17 (01) ◽  
pp. 2050010 ◽  
Author(s):  
Youness Chatibi ◽  
El Hassan El Kinani ◽  
Abdelaziz Ouhadan
Keyword(s):  
Conservation Laws ◽  
Symmetry Algebra ◽  
Lie Symmetry ◽  
Group Method ◽  
Subspace Method ◽  
Lie Symmetry Analysis ◽  
Lie Group Method ◽  
Black Scholes ◽  
The Family ◽  
Symmetry Generators

In this paper, the Lie symmetry algebra admitted by the time fractional Black–Scholes equation is obtained by using the Lie group method. The constructed symmetry generators are investigated to construct a family of exact solutions and conservation laws for the studied equation. At the same time, the family of solutions is extended by using the invariant subspace method.

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