Scattering and Trapping of Nonlinear Schroedinger Solitons in External Potentials

Soliton motion in some external potentials is studied using the nonlinear Schr\"odinger equation. Solitons are scattered by a potential wall. Solitons propagate almost freely or are trapped in a periodic potential. The critical kinetic energy for reflection and trapping is evaluated approximately with a variational method.Comment: 9 pages, 7 figure

Interaction of Nonlinear Schr\"odinger Solitons with an External Potential

Employing a particularly suitable higher order symplectic integration algorithm, we integrate the 1-$d$ nonlinear Schr\"odinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential. We find that the soliton can break up into two solitons, eventually accompanied by radiation of non-solitary waves. Reflection coefficients and inelasticities are computed as functions of the height of the potential step and of its steepness.Comment: 14 pages, uuencoded PS-file including 10 figure

A New Monte Carlo Algorithm for Protein Folding

We demonstrate that the recently proposed pruned-enriched Rosenbluth method (P. Grassberger, Phys. Rev. E 56 (1997) 3682) leads to extremely efficient algorithms for the folding of simple model proteins. We test them on several models for lattice heteropolymers, and compare to published Monte Carlo studies. In all cases our algorithms are faster than all previous ones, and in several cases we find new minimal energy states. In addition to ground states, our algorithms give estimates for the partition sum at finite temperatures.Comment: 4 pages, Latex incl. 3 eps-figs., submitted to Phys. Rev. Lett., revised version with changes in the tex

Determination of the exponent gamma for SAWs on the two-dimensional Manhattan lattice

We present a high-statistics Monte Carlo determination of the exponent gamma for self-avoiding walks on a Manhattan lattice in two dimensions. A conservative estimate is \gamma \gtapprox 1.3425(3), in agreement with the universal value 43/32 on regular lattices, but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations. We find strong corrections to scaling that seem to indicate the presence of a non-analytic exponent Delta < 1. If we assume Delta = 11/16 we find gamma = 1.3436(3), where the error is purely statistical.Comment: 24 pages, LaTeX2e, 4 figure

The Origin of the Designability of Protein Structures

We examined what determines the designability of 2-letter codes (H and P) lattice proteins from three points of view. First, whether the native structure is searched within all possible structures or within maximally compact structures. Second, whether the structure of the used lattice is bipartite or not. Third, the effect of the length of the chain, namely, the number of monomers on the chain. We found that the bipartiteness of the lattice structure is not a main factor which determines the designability. Our results suggest that highly designable structures will be found when the length of the chain is sufficiently long to make the hydrophobic core consisting of enough number of monomers.Comment: 17 pages, 2 figure

Decay of Resonance Structure and Trapping Effect in Potential Scattering Problem of Self-Focusing Wave Packet

Potential scattering problems governed by the time-dependent Gross-Pitaevskii equation are investigated numerically for various values of coupling constants. The initial condition is assumed to have the Gaussian-type envelope, which differs from the soliton solution. The potential is chosen to be a box or well type. We estimate the dependences of reflectance and transmittance on the width of the potential and compare these results with those given by the stationary Schr\"odinger equation. We attribute the behaviors of these quantities to the limitation on the width of the nonlinear wave packet. The coupling constant and the width of the potential play an important role in the distribution of the waves appearing in the final state of scattering.Comment: 18 pages, 12 figures; added 2 figure

Protein folding using contact maps

We present the development of the idea to use dynamics in the space of contact maps as a computational approach to the protein folding problem. We first introduce two important technical ingredients, the reconstruction of a three dimensional conformation from a contact map and the Monte Carlo dynamics in contact map space. We then discuss two approximations to the free energy of the contact maps and a method to derive energy parameters based on perceptron learning. Finally we present results, first for predictions based on threading and then for energy minimization of crambin and of a set of 6 immunoglobulins. The main result is that we proved that the two simple approximations we studied for the free energy are not suitable for protein folding. Perspectives are discussed in the last section.Comment: 29 pages, 10 figure

Competition of Mesoscales and Crossover to Tricriticality in Polymer Solutions

We show that the approach to asymptotic fluctuation-induced critical behavior in polymer solutions is governed by a competition between a correlation length diverging at the critical point and an additional mesoscopic length-scale, the radius of gyration. Accurate light-scattering experiments on polystyrene solutions in cyclohexane with polymer molecular weights ranging from 200,000 up to 11.4 million clearly demonstrate a crossover between two universal regimes: a regime with Ising asymptotic critical behavior, where the correlation length prevails, and a regime with tricritical theta-point behavior determined by a mesoscopic polymer-chain length.Comment: 4 pages in RevTeX with 4 figure

Four-dimensional polymer collapse II: Pseudo-First-Order Transition in Interacting Self-avoiding Walks

In earlier work we provided the first evidence that the collapse, or coil-globule, transition of an isolated polymer in solution can be seen in a four-dimensional model. Here we investigate, via Monte Carlo simulations, the canonical lattice model of polymer collapse, namely interacting self-avoiding walks, to show that it not only has a distinct collapse transition at finite temperature but that for any finite polymer length this collapse has many characteristics of a rounded first-order phase transition. However, we also show that there exists a `$\theta$-point' where the polymer behaves in a simple Gaussian manner (which is a critical state), to which these finite-size transition temperatures approach as the polymer length is increased. The resolution of these seemingly incompatible conclusions involves the argument that the first-order-like rounded transition is scaled away in the thermodynamic limit to leave a mean-field second-order transition. Essentially this happens because the finite-size \emph{shift} of the transition is asymptotically much larger than the \emph{width} of the pseudo-transition and the latent heat decays to zero (algebraically) with polymer length. This scenario can be inferred from the application of the theory of Lifshitz, Grosberg and Khokhlov (based upon the framework of Lifshitz) to four dimensions: the conclusions of which were written down some time ago by Khokhlov. In fact it is precisely above the upper critical dimension, which is 3 for this problem, that the theory of Lifshitz may be quantitatively applicable to polymer collapse.Comment: 30 pages, 14 figures included in tex

Flory-Huggins theory for athermal mixtures of hard spheres and larger flexible polymers

A simple analytic theory for mixtures of hard spheres and larger polymers with excluded volume interactions is developed. The mixture is shown to exhibit extensive immiscibility. For large polymers with strong excluded volume interactions, the density of monomers at the critical point for demixing decreases as one over the square root of the length of the polymer, while the density of spheres tends to a constant. This is very different to the behaviour of mixtures of hard spheres and ideal polymers, these mixtures although even less miscible than those with polymers with excluded volume interactions, have a much higher polymer density at the critical point of demixing. The theory applies to the complete range of mixtures of spheres with flexible polymers, from those with strong excluded volume interactions to ideal polymers.Comment: 9 pages, 4 figure