Shih Feng Shieh

A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrödinger equations

We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrodinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly.

THE SPATIAL ENTROPY OF TWO-DIMENSIONAL SUBSHIFTS OF FINITE TYPE

In this paper, two recursive formulas for computing the spatial entropy of two-dimensional subshifts of finite type are given. The exact entropy of a nontrivial example arising in cellular neural networks is obtained by using such formulas. We also establish some general theory concerning the spatial entropy of two-dimensional subshifts of finite type. In particular, we show that if either of the transition matrices is rank-one, then the associated exact entropy can be explicitly obtained. The generalization of our results to higher dimension can be similarly obtained. Furthermore, these formulas can be used numerically for estimating the spatial entropy.

CELLULAR NEURAL NETWORKS: SPACE-DEPENDENT TEMPLATE, MOSAIC PATTERNS AND SPATIAL CHAOS

We consider a Cellular Neural Network (CNN) with a bias term in the integer lattice tenpoint ℤ2 on the plane tenpoint ℤ2. We impose a space-dependent coupling (template) appropriate for CNN in the hexagonal lattice on tenpoint ℤ2. Stable mosaic patterns of such CNN are completely characterized. The spatial entropy of a tenpoint (p1, p2)-translation invariant set is proved to be well-defined and exists. Using such a theorem, we are also able to address the complexities of resulting mosaic patterns.

Eigenvalue Problems for One-Dimensional Discrete Schrödinger Operators with Symmetric Boundary Conditions

In this paper, we investigate the one-dimensional discrete Schrodinger equation with general, symmetric boundary conditions. Our results primarily concern the number of energy states lying in the wells.

Structure-Preserving Flows of Symplectic Matrix Pairs

We construct a nonlinear differential equation of matrix pairs

A structure-preserving curve for symplectic pairs and its applications

(\mathcal{M}(t),\mathcal{L}(t))

Rotational quotient procedure: A tracking control continuation method for PDEs on radially symmetric domains

that are invariant (structure-preserving property) in the class of symplectic matrix pairs

ON THE SPATIAL ENTROPY OF TWO-DIMENSIONAL GOLDEN MEAN

\left\{ \left(\mathcal{M},\mathcal{L}\right)= \scriptstyle\left.\left(\left[ \begin{smallmatrix} X_{12} && 0 \\ X_{22} && I \\ \end{smallmatrix} \right]\mathcal{S}_2, \left[ \begin{smallmatrix} I && X_{11} \\ 0 && X_{21} \\ \end{smallmatrix} \right]\mathcal{S}_1\right) \right| \ X=[X_{ij}]_{1\le i,j\le2}\ \textstyle{\rm is\ Hermitian} \right\},

Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate

where

A continuation BSOR-Lanczos-Galerkin method for positive bound states of a multi-component Bose-Einstein condensate

\mathcal{S}_1