Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schrödinger Equation

2016 ◽  
Vol 26 (08) ◽  
pp. 1650136 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Invariant Manifolds ◽  
First Integrals ◽  
Phase Portraits ◽  
Planar Dynamical System ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Dimensional Integral

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity

2017 ◽  
Vol 27 (12) ◽  
pp. 1750188
Author(s):  
Yan Zhou ◽  
Jibin Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Optical Waveguides ◽  
Bifurcation Theory ◽  
Phase Portraits ◽  
Explicit Solutions ◽  
Theory Method ◽  
Planar Dynamical System ◽  
Soliton Model ◽  
Straight Lines

Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law nonlinearity is investigated by the method of dynamical systems. The functions [Formula: see text] in the solutions [Formula: see text] [Formula: see text] satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of [Formula: see text], under 23 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 92 exact explicit solutions of system (6) are derived.

Bifurcations and Exact Solutions of the Equation of Barotropic FRW Cosmologies

2017 ◽  
Vol 27 (05) ◽  
pp. 1750080 ◽  
Author(s):  
Jibin Li ◽  
Tonghua Zhang
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Exact Solutions ◽  
Periodic Solutions ◽  
Phase Portraits ◽  
Planar Dynamical System ◽  
Heteroclinic Solutions ◽  
Unbounded Solutions ◽  
Level Curves ◽  
Friedmann Robertson Walker

In this paper, we study the equation of barotropic Friedmann–Robertson–Walker cosmologies. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system. Corresponding to different level curves, we derive exact explicit parametric representations of bounded and unbounded solutions, such as periodic solutions, periodic peakon solutions, homoclinic and heteroclinic solutions and compacton solutions.

DYNAMICAL SYSTEMS AND TESSELATIONS: DETECTING DETERMINISM IN DATA

1991 ◽  
Vol 01 (04) ◽  
pp. 777-794 ◽  
Author(s):  
ALISTAIR I. MEES
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Dynamical Systems ◽  
State Space ◽  
Scientific Theory ◽  
Invariant Manifolds ◽  
Geometric Information ◽  
Topological Information ◽  
Low Dimensional ◽  
Dynamical Information

Data measurements from a dynamical system may be used to build triangulations and tesselations which can — at least when the system has relatively low-dimensional attractors or invariant manifolds — give topological, geometric and dynamical information about the system. The data may consist of a time series, possibly reconstructed by embedding, or of several such series; transients can be put to good use. The topological information which can be found includes dimension and genus of a manifold containing the state space. Geometric information includes information about folds, branches and other chaos generators. Dynamical information is obtained by using the tesselation to construct a map with stated smoothness properties and having the same dynamics as the data; the resulting dynamical model may be tested in the way that any scientific theory may be tested, by making falsifiable predictions.

scholarly journals Periodic Peakon and Smooth Periodic Solutions for KP-MEW(3,2) Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Liping He
Keyword(s):  
Dynamical System ◽  
Periodic Solution ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Phase Portrait ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Wave System ◽  
Integral Constant ◽  
Straight Line

In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for c < 0 , 0 < c < 1 , and c > 1 is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.

scholarly journals Explicit Wave Solutions and Qualitative Analysis of the (1 + 2)-Dimensional Nonlinear Schrödinger Equation with Dual-Power Law Nonlinearity

2015 ◽  
Vol 2015 ◽  
pp. 1-16
Author(s):  
Qing Meng ◽  
Bin He ◽  
Zhenyang Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Power Law ◽  
Bifurcation Theory ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Point Of View ◽  
Phase Portraits ◽  
Nonlinear Schrödinger ◽  
Dual Power

The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is studied using the factorization technique, bifurcation theory of dynamical system, and phase portraits analysis. From a dynamic point of view, the existence of smooth solitary wave, and kink and antikink waves is proved and all possible explicit parametric representations of these waves are presented.

scholarly journals Periodic Wave Solutions and Their Limit Forms of the Modified Novikov Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical System ◽  
Nonlinear Waves ◽  
Bifurcation Theory ◽  
Dynamic Properties ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Novikov Equation

The modified Novikov equationut-utxx+(b+1)u2ux=buuxuxx+u2uxxxis studied by using the bifurcation theory of dynamical system and the method of phase portraits analysis. The existences, dynamic properties, and limit forms of periodic wave solutions forbbeing a negative even are investigated. All possible exact parametric representations of the different kinds of nonlinear waves also are presented.

scholarly journals Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Anti-cubic Nonlinearity

2021 ◽  
Author(s):  
Guoan Xu ◽  
Jibin Li ◽  
Yi Zhang
Keyword(s):  
Dynamical System ◽  
Optical Waveguides ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Phase Portraits ◽  
Wave Transformation ◽  
Cubic Nonlinearity ◽  
Planar Dynamical System ◽  
Soliton Model ◽  
Wave Solutions

Abstract This paper investigates Raman soliton model in optical metamaterials, having anti-cubic nonlinearity. By travelling wave transformation, the model is transformed into a singular planar dynamical system having three singular straight lines. Using the bifurcation theory method of dynamical systems, under different parameter conditions, bifurcations of phase portraits are studied. More than 30 exact explicit solutions of planar dynamical system are derived, such as exact periodic wave solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic peakons and peakons as well as compacton solutions. In more general parametric conditions, all possible solutions are found.

BIANCHI COSMOLOGIES AS DYNAMICAL SYSTEMS WITHOUT THE CONSTRAINT

2000 ◽  
Vol 15 (17) ◽  
pp. 2771-2791
Author(s):  
MAREK SZYDŁOWSKI ◽  
ADAM KRAWIEC
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Phase Portraits ◽  
Finite Domain ◽  
Hamiltonian Constraint ◽  
Two Dimensional ◽  
Class A ◽  
Dimensional Dynamical System ◽  
Analysis Of Dynamics

The Bianchi class A cosmology is treated as a nonlinear dynamical system. In the new variables in which Hamiltonian constraint is solved algebraically, the Bianchi class A model assumes the form of autonomous dynamical system in ℝ4 with polynomial form of vector field. It is proposed that the dimension of minimum reduced phase spaces of unconstrained autonomous systems be treated as a measure of generality of solution. The behavior of these models is studied in terms of qualitative analysis of differential equations. It is shown that the more general Bianchi IX and Bianchi VIII models (called Mixmaster models) can be presented as four-dimensional. We argue that the reduced Mixmaster dynamical systems are chaotic in the same sense as the original ones. The Bianchi I and Bianchi II world models are described by one-dimensional and two-dimensional systems, respectively. We also study dynamics of Bianchi VI0 and Bianchi VII0 models as a three-dimensional dynamical system. For two-dimensional dynamical system, the phase portraits are constructed with the Poincaré sphere which allows the analysis of dynamics both in finite domain and at infinity. For the last class of models we find an invariant submanifold on which systems are analyzed in details.

First Integrals of Discrete System Based on the Principle of Jourdain

2012 ◽  
Vol 479-481 ◽  
pp. 711-714 ◽  
Author(s):  
Ning Zhang ◽  
Gang Ling Zhao
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Discrete System ◽  
Discrete Dynamical System ◽  
Variational Formula ◽  
Discrete Analog ◽  
Discrete Dynamical Systems ◽  
First Integrals ◽  
Discrete Transformation

In this paper, we investigate first integrals of discrete dynamical systems with the variational principle of Jourdain. The operators of discrete transformation are introduced for the system. Based on the Jourdainian generalized variational formula, we derive the discrete analog of Noether-type identity, and then we obtain the first integrals of discrete dynamical system. We discuss an example to illustrate these results.

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