scholarly journals Lagrangian Curve Flows on Symplectic Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 298
Author(s):  
Chuu-Lian Terng ◽  
Zhiwei Wu
Keyword(s):  
Soliton Solutions ◽  
Differential Invariants ◽  
Symplectic Space ◽  
Hamiltonian Structures ◽  
Linearly Independent ◽  
Lagrangian Subspace ◽  
Curve Flows ◽  
Complete Set ◽  
Smooth Map ◽  
Darboux Transforms

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and the span of γ,γx,…,γx(n−1) is a Lagrangian subspace of R2n. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R2n with respect to the symplectic group Sp(2n), (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s C^n(1)-KdV flows and A^2n−1(2)-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.

TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES

2011 ◽  
Vol 25 (19) ◽  
pp. 2637-2656
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
WEI JIANG
Keyword(s):  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Soliton Solutions ◽  
Loop Algebra ◽  
Darboux Transformations ◽  
Soliton Equation ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Loop Algebras ◽  
Soliton Hierarchy

Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

scholarly journals THE sAKNS HIERARCHY

1998 ◽  
Vol 13 (15) ◽  
pp. 1185-1199 ◽  
Author(s):  
HENRIK ARATYN ◽  
ASHOK DAS
Keyword(s):  
Jacobi Identity ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Lax Representation ◽  
System Of Equations ◽  
Open Problems ◽  
Hamiltonian Structures ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Lax Operator

We study, systematically, the properties of the supersymmetric AKNS (sAKNS) hierarchy. In particular, we discuss the Lax representation in terms of a bosonic Lax operator and some special features of the equations and construct the bosonic local charges as well as the fermionic nonlocal charges associated with the system starting from the Lax operator. We obtain the Hamiltonian structures of the system and check the Jacobi identity through the method of prolongation. We also show that this hierarchy of equations can equivalently be described in terms of a fermionic Lax operator. We obtain the zero curvature formulation as well as the conserved charges of the system starting from this fermionic Lax operator which suggests a connection between the two. Finally, starting from the fermionic description of the system, we construct the soliton solutions for this system of equations through Darboux–Bäcklund transformations and describe some open problems.

scholarly journals Continuous and discrete symmetries of renormalization group equations for neutrino oscillations in matter

2021 ◽  
Author(s):  
Shun Zhou
Keyword(s):  
Differential Equations ◽  
Neutrino Oscillations ◽  
Discrete Symmetry ◽  
Differential Invariants ◽  
Neutrino Masses ◽  
Renormalization Group Equations ◽  
Mixing Matrix ◽  
Complete Set ◽  
Independent Variable ◽  
Continuous Symmetries

Abstract Three-flavor neutrino oscillations in matter can be described by three effective neutrino masses mi (for i = 1, 2, 3) and the effective mixing matrix Vαi (for α = e, µ, τ and i = 1, 2, 3). When the matter parameter a ≡ 2√2GFNeE is taken as an independent variable, a complete set of first-order ordinary differential equations for m2 i and |Vαi|2have been derived in the previous works. In the present paper, we point out that such a system of differential equations possesses both the continuous symmetries characterized by one-parameter Lie groups and the discrete symmetry associated with the permutations of three neutrino mass eigenstates. The implications of these symmetries for solving the differential equations and looking for differential invariants are discussed.

scholarly journals Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Muhammad Ayub ◽  
Masood Khan ◽  
F. M. Mahomed
Keyword(s):  
Lie Algebras ◽  
Kepler Problem ◽  
Natural Extension ◽  
Three Dimensional ◽  
Second Order ◽  
Differential Invariants ◽  
Canonical Forms ◽  
Solvable Lie Algebra ◽  
Complete Set ◽  
Invariant Representations

We present a systematic procedure for the determination of a complete set ofkth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of twokth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case ofk= 2 and 31 classes for the case ofk≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of twokth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

scholarly journals CANONICAL QUANTIZATION OF THE BELINSKIĬ-ZAKHAROV ONE-SOLITON SOLUTIONS

1995 ◽  
Vol 04 (06) ◽  
pp. 749-766
Author(s):  
NENAD MANOJLOVIC ◽  
GUILLERMO A. MENA MARUGÁN
Keyword(s):  
Phase Space ◽  
Quantum Dynamics ◽  
Canonical Quantization ◽  
Soliton Solutions ◽  
Irreducible Unitary Representation ◽  
Einstein Equations ◽  
Classical Solutions ◽  
Hamiltonian Constraint ◽  
Dynamical Equations ◽  
Complete Set

We apply the algebraic quantization programme proposed by Ashtekar to the analysis of the Belinskiĭ-Zakharov classical spacetimes, obtained from the Kasner metrics by means of a generalized soliton transformation. When the solitonic parameters associated with this transformation are frozen, the resulting Belinskiĭ-Zakharov metrics provide the set of classical solutions to a gravitational minisuperspace model whose Einstein equations reduce to the dynamical equations generated by a homogeneous Hamiltonian constraint and to a couple of second-class constraints. The reduced phase space of such a model has the symplectic structure of the cotangent bundle over R+×R+. In this reduced phase space, we find a complete set of real observables which form a Lie algebra under Poisson brackets. The quantization of the gravitational model is then carried out by constructing an irreducible unitary representation of that algebra of observables. Finally, we show that the quantum theory obtained in this way is unitarily equivalent to that which describes the quantum dynamics of the Kasner model.

scholarly journals On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy

2006 ◽  
Vol 61 (1-2) ◽  
pp. 7-15
Author(s):  
Amitava Choudhuri ◽  
Benoy Talukdar ◽  
S. B. Dattab
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Hamiltonian Structures ◽  
Lagrangian Equations ◽  
Integrable Evolution Equation ◽  
Integrable Evolution ◽  
Homogeneous Balance Method ◽  
Fifth Order ◽  
Homogeneous Balance

A general form of a fifth-order nonlinear evolution equation is considered. The Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type recursion operator is then employed to construct a hierarchy of Lagrangian equations. It is explicitly demonstrated that the constructed system of equations has a Lax representation and two compatible Hamiltonian structures. The homogeneous balance method is used to derive analytic soliton solutions of the third- and fifth-order equations. - PACS numbers: 47.20.Ky, 42.81.Dp, 02.30.Jr

Stability Analysis of Magnetohydrodynamics Waves in Compressible Turbulent Plasma

2020 ◽  
Vol 9 (3) ◽  
pp. 196-202
Author(s):  
Adel M. Morad ◽  
S. M. A. Maize ◽  
A. A. Nowaya ◽  
Y. S. Rammah
Keyword(s):  
Magnetic Field ◽  
Solitary Waves ◽  
Soliton Solutions ◽  
Reductive Perturbation Method ◽  
Mhd Equations ◽  
Present System ◽  
Stability And Instability ◽  
Regions Of Stability ◽  
Complete Set ◽  
The Magnetic Field

Here, the solitary waves propagation in the cold plasma and their stability conditions are studied. The governing equations are expanded by using the reductive perturbation method with taken under the influence of the magnetic field under consideration. A new nonlinear wave equation is obtained that reconciles the derivative nonlinear Schrdinger equation with a modified form. By considering the magnetic field is constant along the x-direction, the complete set of equations is obtained, and the stable solitary waves are observed. A compariso between the soliton solutions of the modified nonlinear Schrdinger evolution equation (MNLS) and the solutions of the compressible magnetohydrodynamic (MHD) equations has been performed. It is shown that stable solitons can be created in such nonrelativistic fluids in the presence of the magnetic field. The modulation instability for a one-dimensional MNLS equation is carried as well. The regions of stability and instability fo the present system are well determined.

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