Bose-Einstein condensation (BEC) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system, and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum-statistical. The study of BEC in weakly interacting systems which can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.

Bose-Einstein condensation in a gas of sodium atoms

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

Solitons in nonlinear lattices

When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Ei…

Bose-Einstein condensation

We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrodinger equations. The sign of coupling constants β
 ij
 ’s is crucial for the existence of ground state solutions. When all β
 ij
 ’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all β
 ij
 ’s are negative, or one of β
 ij
 ’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N=3.

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Ground State of N Coupled Nonlinear Schrodinger Equations in {\mathbb{R}}^n, n \leq 3

The flow of current through an ionic channel is studied using the energetic variational approach of Liu applied to the primitive (implicit solvent) model of ionic solutions. This approach allows the derivation of self-consistent (Euler–Lagrange) equations to describe the flow of spheres through channels. The partial differential equations derived involve the global interactions of the spheres and are replaced here with a local approximation that we call steric PNP (Poisson–Nernst–Planck) (Lin, T. C.; Eisenberg, B. To be submitted for publication, 2012). Kong combining rules are used and a range of values of steric interaction parameters are studied. These parameters change the energetics of steric interaction but have no effect on diffusion coefficients in models and simulations. Calculations are made for the calcium (EEEE, EEEA) and sodium channels (DEKA) previously studied in Monte Carlo simulations with comparable results. The biological function is quite sensitive to the steric interaction parameters,...

PNP equations with Steric Effects: a Model of Ion Flow through Channels

Here we study the interaction and the configuration of spikes in a double condensate by analyzing least energy solutions of two coupled nonlinear Schrodinger equations which model Bose–Einstein condensates of two different hyperfine spin states. When the interspecies scattering length is positive and large enough, spikes of a double condensate repel each other and behave like two separate spikes. In contrast, spikes of a double condensate attract each other and behave like a single spike if the interspecies scattering length is negative and large enough. Our mathematical arguments can prove such physical phenomena. We first use Nehari’s manifold to construct least energy solutions, and then use some techniques of singular perturbation problems to derive the asymptotic behavior of least energy solutions. It is shown that the interaction term determines the locations of the two spikes and the asymptotic shape of least energy solutions.  2005 Elsevier SAS. All rights reserved. Resume Nous etudions l’interaction et la disposition des pics pour un condensat double. Notre methode utilise l’analyse des solutions avec energie minimale des deux equations de Schrodinger couplees non lineaires qui modelent le condensat de Bose et Einstein pour deux etats des spins differents hyperfinis. Si la longueur de dispersion entre especes est positive et suffisamment gande, les pics pour un condensat double se repoussent, et se comportent comme deux pics separes. En revanche pour un condensat double s’attirent et se comportent comme un seul pic ; la longueur de dispersion entre especes est negative et suffisamment petite. Nos arguments mathematiques etablissent rigoureusement ses phenomenes physiques. Nous utilisons d’une part la variete de Nehari pour la construction de solutions avec energie minimale et d’autre part les techniques de problemes de perturbation singuliere pour la derivation du comportement asymptotique de la solutions avec energie minimale. Nous demontrons que le terme d’interaction determine la position des pics et la forme asymptotique des solutions avec energie minimale.

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Spikes in two coupled nonlinear Schrödinger equations

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Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates

This article proposes a novel hydraulic inerter and experimentally verifies its properties. The inerter is a true two-terminal mechanical network element, which was invented to substitute for the m...

Tai-Chia Lin

Papers

Ground State of N Coupled Nonlinear Schrodinger Equations in {\mathbb{R}}^n, n \leq 3

PNP equations with Steric Effects: a Model of Ion Flow through Channels

Spikes in two coupled nonlinear Schrödinger equations

Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates

Designing and testing a hydraulic inerter