scholarly journals Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Chao Yue ◽  
Tiecheng Xia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Theta Function ◽  
Meromorphic Functions ◽  
Reaction Diffusion ◽  
Arithmetic Genus ◽  
Trigonal Curve ◽  
Matrix Spectral Problem ◽  
Geometric Solutions

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a 3 × 3 matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve K m − 2 of arithmetic genus m − 2 , from which the corresponding Baker-Akhiezer function and meromorphic functions on K m − 2 are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.

scholarly journals New Neumann System Associated with a 3 × 3 Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Fang Li ◽  
Liping Lu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Completely Integrable ◽  
Neumann System ◽  
Matrix Spectral Problem

The nonlinearization approach of Lax pair is applied to the case of the Neumann constraint associated with a 3 × 3 matrix spectral problem, from which a new Neumann system is deduced and proved to be completely integrable in the Liouville sense. As an application, solutions of the first nontrivial equation related to the 3 × 3 matrix spectral problem are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.

A hierarchy of generalized Jaulent–Miodek equations and their explicit solutions

2017 ◽  
Vol 15 (01) ◽  
pp. 1850002
Author(s):  
Xianguo Geng ◽  
Liang Guan ◽  
Bo Xue
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Spectral Problem ◽  
Theta Function ◽  
Algebraic Curves ◽  
Explicit Solutions ◽  
Two Systems ◽  
Energy Dependent

A hierarchy of generalized Jaulent–Miodek (JM) equations related to a new spectral problem with energy-dependent potentials is proposed. Depending on the Lax matrix and elliptic variables, the generalized JM hierarchy is decomposed into two systems of solvable ordinary differential equations. Explicit theta function representations of the meromorphic function and the Baker–Akhiezer function are constructed, the solutions of the hierarchy are obtained based on the theory of algebraic curves.

Algebro-geometric integration of a modified shallow wave hierarchy

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Guoliang He ◽  
Yunyun Zhai ◽  
Zhenzhen Zheng
Keyword(s):  
Wave Equations ◽  
Meromorphic Functions ◽  
Asymptotic Properties ◽  
Trigonal Curve ◽  
Recursion Relations ◽  
Curvature Equation ◽  
The Third ◽  
Zero Curvature ◽  
Matrix Spectral Problem ◽  
Geometric Solutions

Abstract By introducing two sets of Lenard recursion relations, we derive a hierarchy of modified shallow wave equations associated with a 3 × 3 matrix spectral problem with three potentials from the zero-curvature equation. The Baker–Akhiezer function and two meromorphic functions are defined on the trigonal curve which is introduced by utilizing the characteristic polynomial of the Lax matrix. Analyzing the asymptotic properties of the Baker–Akhiezer function and two meromorphic functions at two infinite points, we arrive at the explicit algebro-geometric solutions for the entire hierarchy in terms of the Riemann theta function by showing the explicit forms of the normalized Abelian differentials of the third kind.

TRANSITION WAVES THAT LEAVE BEHIND REGULAR OR IRREGULAR SPATIOTEMPORAL OSCILLATIONS IN A SYSTEM OF THREE REACTION–DIFFUSION EQUATIONS

1993 ◽  
Vol 03 (05) ◽  
pp. 1269-1279 ◽  
Author(s):  
JONATHAN A. SHERRATT
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Wave Front ◽  
Plane Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spatiotemporal Oscillations ◽  
Transition Wave ◽  
Periodic Plane

Transition waves are widespread in the biological and chemical sciences, and have often been successfully modelled using reaction–diffusion systems. I consider a particular system of three reaction–diffusion equations, and I show that transition waves can destabilise as the kinetic ordinary differential equations pass through a Hopf bifurcation, giving rise to either regular or irregular spatiotemporal oscillations behind the advancing transition wave front. In the case of regular oscillations, I show that these are periodic plane waves that are induced by the way in which the transition wave front approaches its terminal steady state. Further, I show that irregular oscillations arise when these periodic plane waves are unstable as reaction–diffusion solutions. The resulting behavior is not related to any chaos in the kinetic ordinary differential equations.

Quasi-periodic Solutions to the K(−2, −2) Hierarchy

2016 ◽  
Vol 71 (7) ◽  
pp. 639-645
Author(s):  
Lihua Wu ◽  
Xianguo Geng
Keyword(s):  
Meromorphic Function ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Characteristic Polynomial ◽  
Theta Function ◽  
Hyperelliptic Curve ◽  
Arithmetic Genus ◽  
Riemann Theta Function ◽  
Function Representation

AbstractWith the help of the characteristic polynomial of Lax matrix for the K(−2, −2) hierarchy, we define a hyperelliptic curve 𝒦n+1 of arithmetic genus n+1. By introducing the Baker–Akhiezer function and meromorphic function, the K(−2, −2) hierarchy is decomposed into Dubrovin-type differential equations. Based on the theory of hyperelliptic curve, the explicit Riemann theta function representation of meromorphic function is given, and from which the quasi-periodic solutions to the K(−2, −2) hierarchy are obtained.

scholarly journals Numerical Simulation and Symmetry Reduction of a Two-Component Reaction-Diffusion System

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jina Li ◽  
Xuehui Ji
Keyword(s):  
Numerical Simulation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Reaction Diffusion ◽  
Symmetry Reduction ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Symmetry Classification ◽  
Two Component

In this paper, the symmetry classification and symmetry reduction of a two-component reaction-diffusion system are investigated, the reaction-diffusion system can be reduced to system of ordinary differential equations, and the solutions and numerical simulation will be showed by examples.

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