scholarly journals Generalised Sasa–Satsuma Equation: Densities Approach to New Infinite Hierarchy of Integrable Evolution Equations

2018 ◽  
Vol 73 (12) ◽  
pp. 1121-1128 ◽  
Author(s):  
A. Ankiewicz ◽  
U. Bandelow ◽  
N. Akhmediev
Keyword(s):  
Dynamical System ◽  
Time Evolution ◽  
Evolution Equations ◽  
Lax Pair ◽  
High Order ◽  
Infinite Hierarchy ◽  
Evolution Pattern ◽  
Integrable Evolution ◽  
Invariant Densities

AbstractWe derive the new infinite Sasa–Satsuma hierarchy of evolution equations using an invariant densities approach. Being significantly simpler than the Lax-pair technique, this approach does not involve ponderous 3 × 3 matrices. Moreover, it allows us to explicitly obtain operators of many orders involved in the time evolution of the Sasa–Satsuma hierarchy functionals. All these operators are parts of a generalised Sasa–Satsuma equation of infinitely high order. They enter this equation with independent arbitrary real coefficients that govern the evolution pattern of this multiparameter dynamical system.

Integrability aspects and some abundant solutions for a new (4 + 1)-dimensional KdV-like equation

2021 ◽  
pp. 2150079
Author(s):  
Peng-Fei Han ◽  
Taogetusang Bao
Keyword(s):  
Evolution Equations ◽  
Dynamic Models ◽  
Dynamic Properties ◽  
Soliton Solutions ◽  
Lax Pair ◽  
High Order ◽  
Nonlinear Evolution Equations ◽  
Polynomial Method ◽  
Bilinear Method ◽  
Text Type

In this paper, we introduce a new (4 + 1)-dimensional KdV-like equation. By using the Bell Polynomial method, we obtain the bilinear form, bilinear Bäcklund transformation, Lax pair and infinite conservation laws. It is proved that the equation is completely integrable in Lax pair sense. Based on the Hirota bilinear method and the test function method, high-order lump solutions, high-order lump-kink type [Formula: see text]-soliton solutions, high-order lump-[Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text] type soliton solutions, [Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text] type soliton solutions and [Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text]-[Formula: see text] type soliton solutions [Formula: see text] for this equation are obtained with the help of symbolic computation. Via three-dimensional plots and contour plots with the help of Mathematics, analyses for the obtained solutions are presented, and their dynamic properties are discussed. Many dynamic models can be simulated by nonlinear evolution equations, and these graphical analyses are helpful to understand these models.

Motion of an integral curve of a Hamiltonian dynamical system and the evolution equations in 3D

2017 ◽  
Vol 14 (12) ◽  
pp. 1750172
Author(s):  
T. Bayrakdar ◽  
A. A. Ergin
Keyword(s):  
Dynamical System ◽  
Time Evolution ◽  
Poisson Structure ◽  
Evolution Equations ◽  
Integral Curve ◽  
Mkdv Equation ◽  
Three Dimensions ◽  
Geodesic Curve ◽  
Hamiltonian Dynamical System ◽  
De Vries

We show that all of the nonstretching curve motions specified in the Frenet–Serret frame in the literature can be described by the time evolution of an integral curve of a Hamiltonian dynamical system such that the underlying curve is a geodesic curve on a leaf of the foliation determined by the Poisson structure in three dimensions. As an illustrative example, we show that the focusing version of the nonlinear Schrödinger equation and the complex modified Korteweg–de Vries (mKdV) equation are obtained in this way.

A HIERARCHY OF HAMILTONIAN EQUATIONS ASSOCIATED WITH THE MODIFIED JAULENT–MIODEK HIERARCHY

2010 ◽  
Vol 24 (02) ◽  
pp. 183-193
Author(s):  
HAI-YONG DING ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN ◽  
LI-LI ZHU
Keyword(s):  
Eigenvalue Problem ◽  
Evolution Equations ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Matrix Eigenvalue ◽  
Integrable Evolution

By considering a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is presented.

scholarly journals Dynamical system analysis of three-form field dark energy model with baryonic matter

2020 ◽  
Vol 80 (9) ◽  
Author(s):  
Soumya Chakraborty ◽  
Sudip Mishra ◽  
Subenoy Chakraborty
Keyword(s):  
Dynamical System ◽  
Dark Energy ◽  
Cold Dark Matter ◽  
System Analysis ◽  
Evolution Equations ◽  
Matter Field ◽  
Baryonic Matter ◽  
Form Field ◽  
The Stability ◽  
Manifold Theory

AbstractA cosmological model having matter field as (non) interacting dark energy (DE) and baryonic matter and minimally coupled to gravity is considered in the present work with flat FLRW space time. The DE is chosen in the form of a three-form field while radiation and dust (i.e; cold dark matter) are the baryonic part. The cosmic evolution is studied through dynamical system analysis of the autonomous system so formed from the evolution equations by suitable choice of the dimensionless variables. The stability of the non-hyperbolic critical points are examined by Center manifold theory and possible bifurcation scenarios have been examined.

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