Algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations via Hamiltonian approach

2015 ◽  
Vol 12 (03) ◽  
pp. 1550038
Author(s):  
Dianlou Du ◽  
Xiao Yang
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Equations ◽  
Poisson Structure ◽  
Hamiltonian Approach ◽  
Nonlinear Schrödinger ◽  
Hamiltonian Flows ◽  
Jacobi Inversion ◽  
Liouville Integrable ◽  
Separated Variables

The algebraic-geometrical solutions of three (2 + 1)-dimensional equations (including mKP equation and coupled mKP equation) are discussed by Hamiltonian approach. First, the Poisson structure on CN × RN is introduced to give a Hamiltonian system associated with the derivative nonlinear Schrödinger (DNLS) hierarchy. The Hamiltonian system is proved to be Liouville integrable, accordingly the solutions of three (2 + 1)-dimensional nonlinear equations can be solved by three compatible Hamiltonian flows. Second, the canonical separated variables and Hamilton–Jacobi theory is used to definite action-angle variables for Hamiltonian flows. At last, by Riemann–Jacobi inversion, the algebraic-geometrical solutions of three (2 + 1)-dimensional nonlinear equations are obtained. Besides, the algebraic-geometrical solutions of the first two DNLS equations are also given.

scholarly journals Hamiltonian Approach to Magnetic Fields with Toroidal Surfaces

1985 ◽  
Vol 40 (10) ◽  
pp. 959-967
Author(s):  
A. Salat
Keyword(s):  
Magnetic Field ◽  
Hamiltonian System ◽  
Magnetic Fields ◽  
Constant Pressure ◽  
Time Dependent ◽  
Hamiltonian Approach ◽  
One Dimensional ◽  
Mhd Equilibrium ◽  
Magnetic Surfaces ◽  
Rotational Transform

The equivalence of magnetic field line equations to a one-dimensional time-dependent Hamiltonian system is used to construct magnetic fields with arbitrary toroidal magnetic surfaces I = const. For this purpose Hamiltonians H which together with their invariants satisfy periodicity constraints have to be known. The choice of H fixes the rotational transform η(I). Arbitrary axisymmetric fields, and nonaxisymmetric fields with constant η(I) are considered in detail.Configurations with coinciding magnetic and current density surfaces are obtained. The approach used is not well suited, however, to satisfying the additional MHD equilibrium condition of constant pressure on magnetic surfaces.

scholarly journals The soliton solution of a modified nonlinear schrödinger equation

2017 ◽  
Vol 5 (1) ◽  
pp. 16
Author(s):  
Jumei Zhang ◽  
Li Yin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Equations ◽  
Nonlinear Schrödinger Equation ◽  
Soliton Solution ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Derivative Method ◽  
Hirota Bilinear

Hirota bilinear derivative method can be used to construct the soliton solutions for nonlinear equations. In this paper we construct the soliton solutions of a modified nonlinear Schrödinger equation by bilinear derivative method.

THE $W^{(2)}_3$ CONFORMAL ALGEBRA AND THE BOUSSINESQ HIERARCHY

1991 ◽  
Vol 06 (26) ◽  
pp. 2397-2409 ◽  
Author(s):  
P. MATHIEU ◽  
W. OEVEL
Keyword(s):  
Nonlinear Equations ◽  
Poisson Structure ◽  
Boussinesq Equation ◽  
Free Field ◽  
Conformal Algebra ◽  
Integrable Hierarchy ◽  
Classical Formula ◽  
Field Representation ◽  
Space And Time ◽  
Time Variables

The classical [Formula: see text] algebra Polyakov is shown to be equivalent to the second Poisson structure of a new integrable hierarchy of nonlinear equations. The hierarchy is related to the Boussinesq hierarchy by interhcanging the roles of the space and time variables x and t in the Boussinesq equation. From this relation the Miura map, relating the new hierarchy to its modified version, can be derived systematically. It is found to be equivalent to the known free field representation of the [Formula: see text] algebra.

Symplectic schemes and symmetric schemes for nonlinear Schrödinger equation in the case of dark solitons motion

2021 ◽  
pp. 2150056
Author(s):  
Yiming Yao ◽  
Miao Xu ◽  
Beibei Zhu ◽  
Quandong Feng
Keyword(s):  
Hamiltonian System ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Dark Soliton ◽  
Nonlinear Schrodinger Equation ◽  
Conserved Quantities ◽  
Coordinate Transformations ◽  
Nonlinear Schrödinger ◽  
Symplectic Schemes

In this paper, symplectic schemes and symmetric schemes are presented to simulate Nonlinear Schrödinger Equation (NLSE) in case of dark soliton motion. Firstly, by Ablowitz–Ladik model (A–L model), the NLSE is discretized into a non-canonical Hamiltonian system. Then, different kinds of coordinate transformations can be used to standardize the non-canonical Hamiltonian system. Therefore, the symplectic schemes and symmetric schemes can be employed to simulate the solitons motion and test the preservation of the invariants of the A–L model and the conserved quantities approximations of the original NLSE. The numerical experiments show that symplectic schemes and symmetric schemes have similar simulation effect, and own significant superiority over non-symplectic and non-symmetric schemes in long-term tracking the motion of solitons, preserving the invariants and the approximations of conserved quantities. Moreover, it is obvious that coordinate transformations with more symmetry have a better simulation effect.

scholarly journals A Hamiltonian Approach to Fault Isolation in a Planar Vertical Take–Off and Landing Aircraft Model

2015 ◽  
Vol 25 (1) ◽  
pp. 65-76 ◽  
Author(s):  
Luis H. Rodriguez-Alfaro ◽  
Efrain Alcorta-Garcia ◽  
David Lara ◽  
Gerardo Romero
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Systems ◽  
Fault Detection ◽  
Fault Isolation ◽  
Fault Detection And Isolation ◽  
System Model ◽  
Actuator Faults ◽  
Hamiltonian Approach ◽  
Aircraft Model ◽  
Fault Sensitivity

Abstract The problem of fault detection and isolation in a class of nonlinear systems having a Hamiltonian representation is considered. In particular, a model of a planar vertical take-off and landing aircraft with sensor and actuator faults is studied. A Hamiltonian representation is derived from an Euler-Lagrange representation of the system model considered. In this form, nonlinear decoupling is applied in order to obtain subsystems with (as much as possible) specific fault sensitivity properties. The resulting decoupled subsystem is represented as a Hamiltonian system and observer-based residual generators are designed. The results are presented through simulations to show the effectiveness of the proposed approach.

Finite Genus Solutions to the Generalised Kaup-Newell Hierarchy and Two 2+1 Dimensional Modified Korteweg-de Vries Equations

2017 ◽  
Vol 72 (7) ◽  
pp. 589-594
Author(s):  
Xiao Yang ◽  
Jiayan Han
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Algebraic Curve ◽  
Inversion Theorem ◽  
First Order ◽  
Hamiltonian Flows ◽  
De Vries ◽  
Jacobi Inversion ◽  
Finite Genus

AbstractA generalised Kaup-Newell (gKN) hierarchy is introduced, which starts with a system of first-order ordinary differential equations and includes the Gerdjikov-Ivanov equation. By introducing an appropriate generating function, its related Hamiltonian systems and algebraic curve are given. The Hamiltonian systems are proved to be integrable, then the gKN hierarchy is solved by Hamiltonian flows. The algebraic curve is provided with suitable genus, then based on the trace formula and Riemann-Jacobi inversion theorem, finite genus solutions of the gKN hierarchy are obtained. Besides, two 2+1 dimensional modified Korteweg-de Vries (mKdV) equations are also solved.

MEASURE SYNCHRONIZATION IN A COUPLED HAMILTONIAN SYSTEM ASSOCIATED WITH NONLINEAR SCHRÖDINGER EQUATION

2005 ◽  
Vol 19 (15) ◽  
pp. 737-742 ◽  
Author(s):  
U. E. VINCENT ◽  
A. N. NJAH ◽  
O. AKINLADE
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Invariant Measure ◽  
Schrödinger Equation ◽  
Hamiltonian Systems ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Measure Synchronization

We present preliminary numerical findings concerning measure synchronization in a pair of coupled Nonlinear Hamiltonian Systems (NLHS) derived from a Nonlinear Schrödinger Equation (NLSE). The dynamics of the two coupled NLHS were found to exhibit a transition to coherent invariant measure; their orbits sharing the same phase space as the coupling strength is increased. Transitions from quasiperiodicity (QP) measure desynchronization to QP measure synchronization and QP measure desynchronization to chaotic (CH) measure synchronization were observed.

scholarly journals The Symplectic Geometry of Polygons in the 3-Sphere

2002 ◽  
Vol 54 (1) ◽  
pp. 30-54 ◽  
Author(s):  
Thomas Treloar
Keyword(s):  
Hamiltonian System ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Integrable Hamiltonian System ◽  
Conjugacy Classes ◽  
Fusion Product ◽  
Hamiltonian Flows ◽  
Theoretic Description

AbstractWe study the symplectic geometry of the moduli spaces Mr = Mr() of closed n-gons with fixed side-lengths in the 3-sphere. We prove that these moduli spaces have symplectic structures obtained by reduction of the fusion product of n conjugacy classes in SU(2) by the diagonal conjugation action of SU(2). Here the fusion product of n conjugacy classes is a Hamiltonian quasi-Poisson SU(2)-manifold in the sense of [AKSM]. An integrable Hamiltonian system is constructed on Mr in which the Hamiltonian flows are given by bending polygons along a maximal collection of nonintersecting diagonals. Finally, we show the symplectic structure on Mr relates to the symplectic structure obtained from gauge-theoretic description of Mr. The results of this paper are analogues for the 3-sphere of results obtained for Mr(), the moduli space of n-gons with fixed side-lengths in hyperbolic 3-space [KMT], and for Mr(), the moduli space of n-gons with fixed side-lengths in [KM1].

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