scholarly journals On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Nina Xue ◽  
Wencai Zhao
Keyword(s):  
Hamiltonian System ◽  
Schrödinger Equation ◽  
Small Parameter ◽  
Hamiltonian Systems ◽  
Schrodinger Equation ◽  
Linear Hamiltonian System ◽  
Constant Matrix ◽  
Change Of Variables ◽  
Multiple Eigenvalues ◽  
Linear Hamiltonian Systems

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian system ẋ=A+εQt, where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as Qt. Applications to the Schrödinger equation are also given.

scholarly journals On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Nina Xue ◽  
Wencai Zhao
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Linear Hamiltonian System ◽  
Constant Matrix ◽  
Change Of Variables ◽  
Periodic Linear ◽  
Multiple Eigenvalues ◽  
Linear Hamiltonian Systems ◽  
Constant Coefficients ◽  
Exponentially Small

In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system x˙=A+εQt,εx, ε∈0,ε0, where A is a constant matrix with possible multiple eigenvalues and Q(t,ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to y˙=A⁎ε+εR⁎t,εy, ε∈0,ε⁎, where R⁎ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.

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 Accurate Spectral Collocation Computations of High Order Eigenvalues for Singular Schrödinger Equations-Revisited

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 761
Author(s):  
Călin-Ioan Gheorghiu
Keyword(s):  
Schrödinger Equation ◽  
Small Parameter ◽  
Error Estimation ◽  
Schrodinger Equation ◽  
Error Control ◽  
High Order ◽  
Order Of Approximation ◽  
Schrödinger Equations ◽  
Spectral Collocation ◽  
Integration Interval

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

QUASIPERIODIC ROUTE TO SPATIOTEMPORAL CHAOS IN A NONLINEAR SCHRÖDINGER EQUATION WITH THE SATURABLE NONLINEARITY

1994 ◽  
Vol 08 (24) ◽  
pp. 1511-1516
Author(s):  
CANGTAO ZHOU
Keyword(s):  
Hamiltonian System ◽  
Schrödinger Equation ◽  
Coherent Structures ◽  
Schrodinger Equation ◽  
Spatiotemporal Chaos ◽  
Nonlinear Schrodinger Equation ◽  
The Self ◽  
Saturable Nonlinearity ◽  
Self Focusing ◽  
Non Linear

The route from the coherent structures to the spatiotemporal complicated patterns is numerically investigated in terms of a continuum Hamiltonian system, that is, the non-linear Schrödinger equation with the self-focusing nonlinearity, where the quasiperiodic route is first observed.

scholarly journals New Interval Oscillation Criteria for Certain Linear Hamiltonian Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jing Shao ◽  
Fanwei Meng
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Linear Matrix ◽  
Riccati Transformation ◽  
Integral Means ◽  
Oscillation Criteria ◽  
General Integral ◽  
Generalized Riccati Transformation ◽  
Linear Hamiltonian Systems ◽  
Interval Oscillation

Using a generalized Riccati transformation and the general integral means technique, some new interval oscillation criteria for the linear matrix Hamiltonian systemU'=(A(t)-λ(t)I)U+B(t)V,V'=C(t)U+(μ(t)I-A*(t))V,t≥t0are obtained. These results generalize and improve the oscillation criteria due to Zheng (2008). An example is given to dwell upon the importance of our results.

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 Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

2018 ◽  
Vol 28 (13) ◽  
pp. 1850168
Author(s):  
Ting Chen ◽  
Jaume Llibre
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Normal Forms ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Linear Change ◽  
Rational Potential ◽  
Change Of Variables ◽  
Poincaré Disk

In this paper, we study the global dynamical behavior of the Hamiltonian system [Formula: see text], [Formula: see text] with the rational potential Hamiltonian [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincaré disk and provide their bifurcation diagrams.

scholarly journals Some Oscillation Results for Linear Hamiltonian Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Nan Wang ◽  
Fanwei Meng
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Riccati Technique ◽  
Averaging Technique ◽  
Oscillation Criteria ◽  
Integral Averaging Technique ◽  
Linear Hamiltonian Systems ◽  
Interval Oscillation ◽  
Generalized Matrix ◽  
Positive Functionals

The purpose of this paper is to develop a generalized matrix Riccati technique for the selfadjoint matrix Hamiltonian systemU′=A(t)U+B(t)V,V′=C(t)U−A∗(t)V. By using the standard integral averaging technique and positive functionals, new oscillation and interval oscillation criteria are established for the system. These criteria extend and improve some results that have been required before. An interesting example is included to illustrate the importance of our results.

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