Quasi-periodic solutions of the Belov–Chaltikian lattice hierarchy

2017 ◽  
Vol 29 (08) ◽  
pp. 1750025 ◽  
Author(s):  
Xianguo Geng ◽  
Xin Zeng
Keyword(s):  
Asymptotic Behavior ◽  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Jacobian Variety ◽  
Trigonal Curve ◽  
Essential Properties ◽  
Discrete Flow ◽  
Abel Map ◽  
Matrix Spectral Problem

Utilizing the characteristic polynomial of Lax matrix for the Belov–Chaltikian (BC) lattice hierarchy associated with a [Formula: see text] discrete matrix spectral problem, we introduce a trigonal curve with three infinite points, from which we establish the associated Dubrovin-type equations. The essential properties of the Baker–Akhiezer function and the meromorphic function are discussed, that include their asymptotic behavior near three infinite points on the trigonal curve and the divisor of the meromorphic function. The Abel map is introduced to straighten out the continuous flow and the discrete flow in the Jacobian variety, from which the quasi-periodic solutions of the entire BC lattice hierarchy are obtained in terms of the Riemann theta function.

scholarly journals QUASI-PERIODIC SOLUTIONS OF THE RELATIVISTIC TODA HIERARCHY

2012 ◽  
Vol 19 (04) ◽  
pp. 1250030 ◽  
Author(s):  
DONG GONG ◽  
XIANGUO GENG
Keyword(s):  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Hyperelliptic Curve ◽  
Algebraic Curves ◽  
Asymptotic Properties ◽  
Toda Hierarchy ◽  
Discrete Flow

On the basis of the theory of algebraic curves, the continuous flow and discrete flow related to the relativistic Toda hierarchy are straightened out using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the relativistic Toda hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

QUASI-PERIODIC SOLUTIONS OF THE DISCRETE mKdV HIERARCHY

2013 ◽  
Vol 10 (03) ◽  
pp. 1250094 ◽  
Author(s):  
XIANGUO GENG ◽  
DONG GONG
Keyword(s):  
Meromorphic Function ◽  
Periodic Solutions ◽  
Continuous Flow ◽  
Hyperelliptic Curve ◽  
Algebraic Curves ◽  
Asymptotic Properties ◽  
Mkdv Hierarchy

On the basis of the theory of algebraic curves, the continuous flow related to the discrete mKdV hierarchy is straightened using the Abel–Jacobi coordinates. The meromorphic function and the Baker–Akhiezer function are introduced on the hyperelliptic curve. Quasi-periodic solutions of the discrete mKdV hierarchy are constructed with the help of the asymptotic properties and the algebro-geometric characters of the meromorphic function and the hyperelliptic curve.

ASYMPTOTIC BEHAVIOR OF SPATIALLY INHOMOGENEOUS BALANCE LAWS

2005 ◽  
Vol 02 (03) ◽  
pp. 645-672 ◽  
Author(s):  
JULIA EHRT ◽  
JÖRG HÄRTERICH
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Initial Data ◽  
Periodic Solutions ◽  
Balance Laws ◽  
Spatially Inhomogeneous ◽  
Periodic Initial Data ◽  
Left And Right ◽  
Longtime Behavior ◽  
Time Periodic

We study the longtime behavior of spatially inhomogeneous scalar balance laws with periodic initial data and a convex flux. Our main result states that for a large class of initial data the entropy solution will either converge uniformly to some steady state or to a discontinuous time-periodic solution. This extends results of Lyberopoulos, Sinestrari and Fan and Hale obtained in the spatially homogeneous case. The proof is based on the method of generalized characteristics together with ideas from dynamical systems theory. A major difficulty consists of finding the periodic solutions which determine the asymptotic behavior. To this end we introduce a new tool, the Rankine–Hugoniot vector field, which describes the motion of a (hypothetical) shock with certain prescribed left and right states. We then show the existence of periodic solutions of the Rankine–Hugoniot vector field and prove that the actual shock curves converge to these periodic solutions.

CHAOTIC ASYMPTOTIC BEHAVIOR OF THE HYPERBOLIC HEAT TRANSFER EQUATION SOLUTIONS

2010 ◽  
Vol 20 (09) ◽  
pp. 2943-2947 ◽  
Author(s):  
J. A. CONEJERO ◽  
A. PERIS ◽  
M. TRUJILLO
Keyword(s):  
Heat Transfer ◽  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Periodic Solutions ◽  
Transfer Equation ◽  
Chaotic Behavior ◽  
Heat Transfer Equation

We describe the periodic solutions and study the chaotic behavior of the solution semigroup of a Cauchy problem corresponding to the hyperbolic heat transfer equation.

scholarly journals On Stability of Periodic Solutions of Lienard Type Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Zijian Yin ◽  
Hongbin Chen
Keyword(s):  
Asymptotic Behavior ◽  
Periodic Solutions ◽  
Exponential Decay ◽  
Floquet Theory ◽  
Linear Growth ◽  
Liénard Equation ◽  
Damping Term ◽  
Restoring Force ◽  
Stable Periodic ◽  
The Stability

We use the Floquet theory to analyze the stability of periodic solutions of Lienard type equations under the asymptotic linear growth of restoring force in this paper. We find that the existence and the stability of periodic solutions are determined primarily by asymptotic behavior of damping term. For special type of Lienard equation, the uniqueness and stability of periodic solutions are obtained. Furthermore, the sharp rate of exponential decay of the stable periodic solutions is determined under suitable conditions imposed on restoring force.

On Algebro-Geometric Solutions of the Camassa-Holm Hierarchy

2007 ◽  
Vol 7 (3) ◽  
Author(s):  
Luca Zampogni
Keyword(s):  
Global Solutions ◽  
Natural Setting ◽  
Jacobian Variety ◽  
Generalized Jacobian ◽  
Zero Curvature ◽  
Geometric Type ◽  
Recursion Formulas ◽  
Abel Map ◽  
Sturm Liouville ◽  
Geometric Solutions

AbstractWe find global solutions of algebro geometric type for all the equations of a new commuting hierarchy containing the Camassa-Holm equation. This hierarchy is built in analogy to the classical K-dV and AKNS hierarchies. We use a zero curvature method to give recursion formulas. The time evolution of the solutions is completely determined, and the motion on a nonlinear subvariety Υ of a generalized Jacobian variety is obtained by solving an inverse problem for the Sturm-Liouville equation L(φ) = −φ″ + φ = λyφ. This is the natural setting for the expression of the solutions which depend linearly with respect to t and x, with coordinates on a curvilinear parallelogram contained in such a subvariety φ. φ is obtained as the restriction of the generalized Abel map I

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