Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations

1995 ◽  
Vol 7 (1) ◽  
pp. 73-107 ◽  
Author(s):  
Xu-Yan Chen ◽  
Peter Polácik
Keyword(s):  
Parabolic Equations ◽  
One Dimensional ◽  
Morse Decompositions ◽  
Time Periodic

scholarly journals MULTIBREATHER AND VORTEX BREATHER STABILITY IN KLEIN–GORDON LATTICES: EQUIVALENCE BETWEEN TWO DIFFERENT APPROACHES

2011 ◽  
Vol 21 (08) ◽  
pp. 2161-2177 ◽  
Author(s):  
J. CUEVAS ◽  
V. KOUKOULOYANNIS ◽  
P. G. KEVREKIDIS ◽  
J. F. R. ARCHILLA
Keyword(s):  
Coupled Oscillators ◽  
Hamiltonian Method ◽  
Band Theory ◽  
Floquet Multipliers ◽  
One Dimensional ◽  
Case Examples ◽  
Discrete Breathers ◽  
Klein Gordon ◽  
Band Method ◽  
Time Periodic

In this work, we revisit the question of stability of multibreather configurations, i.e. discrete breathers with multiple excited sites at the anti-continuum limit of uncoupled oscillators. We present two methods that yield quantitative predictions about the Floquet multipliers of the linear stability analysis around such exponentially localized in space, time-periodic orbits, based on the Aubry band method and the MacKay effective Hamiltonian method, and prove that by making the suitable assumptions about the form of the bands in the Aubry band theory, their conclusions are equivalent. Subsequently, we showcase the usefulness of the methods through a series of case examples including one-dimensional multi-breathers, and two-dimensional vortex breathers in the case of a lattice of linearly coupled oscillators with the Morse potential and in that of the discrete ϕ4 model.

The backward problem of parabolic equations with the measurements on a discrete set

2020 ◽  
Vol 28 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Jin Cheng ◽  
Yufei Ke ◽  
Ting Wei
Keyword(s):  
Parabolic Equations ◽  
Numerical Approximation ◽  
Cross Validation ◽  
Continuation Method ◽  
Measurement Data ◽  
Initial Value ◽  
One Dimensional ◽  
Backward Problem ◽  
The One ◽  
Validation Rule

AbstractThe backward problems of parabolic equations are of interest in the study of both mathematics and engineering. In this paper, we consider a backward problem for the one-dimensional heat conduction equation with the measurements on a discrete set. The uniqueness for recovering the initial value is proved by the analytic continuation method. We discretize this inverse problem by a finite element method to deduce a severely ill-conditioned linear system of algebra equations. In order to overcome the ill-posedness, we apply the discrete Tikhonov regularization with the generalized cross validation rule to obtain a stable numerical approximation to the initial value. Numerical results for three examples are provided to show the effect of the measurement data.

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