Critical Unmixing of Polymer Solutions

We present Monte Carlo simulations of semidilute solutions of long self-attracting chain polymers near their Ising type critical point. The polymers are modeled as monodisperse self-avoiding walks on the simple cubic lattice with attraction between non-bonded nearest neighbors. Chain lengths are up to N=2048, system sizes are up to $2^{21}$ lattice sites and $2.8\times 10^5$ monomers. These simulations used the recently introduced pruned-enriched Rosenbluth method which proved extremely efficient, together with a histogram method for estimating finite size corrections. Our most clear result is that chains at the critical point are Gaussian for large $N$, having end-to-end distances $R\sim\sqrt{N}$. Also the distance $T_\Theta-T_c(N)$ (where $T_\Theta = \lim_{N\to\infty} T_c(N)$) scales with the mean field exponent, $T_\Theta -T_c(N)\sim 1/\sqrt{N}$. The critical density seems to scale with a non-trivial exponent similar to that observed in experiments. But we argue that this is due to large logarithmic corrections. These corrections are similar to the very large corrections to scaling seen in recent analyses of $\Theta$-polymers, and qualitatively predicted by the field theoretic renormalization group. The only serious deviation from this simple global picture concerns the N-dependence of the order parameter amplitudes which disagrees with a minimalistic ansatz of de Gennes. But this might be due to problems with finite size scaling. We find that the finite size dependence of the density of states $P(E,n)$ (where $E$ is the total energy and $n$ is the number of chains) is slightly but significantly different from that proposed recently by several authors.Comment: minor changes; Latex, 22 pages, submitted to J. Chem. Phy

Studies of Phase Turbulence in the One Dimensional Complex Ginzburg-Landau Equation

The phase-turbulent (PT) regime for the one dimensional complex Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large systems and long integration times, using an efficient new integration scheme. Particular attention is paid to solutions with a non-zero phase gradient. For fixed control parameters, solutions with conserved average phase gradient $\nu$ exist only for $|\nu|$ less than some upper limit. The transition from phase to defect-turbulence happens when this limit becomes zero. A Lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient. For high values of the phase gradient a family of non-chaotic solutions of the CGLE is found. These solutions consist of spatially periodic or aperiodic waves travelling with constant velocity. They typically have incommensurate velocities for phase and amplitude propagation, showing thereby a novel type of quasiperiodic behavior. The main features of these travelling wave solutions can be explained through a modified Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the PT phase. The latter explains also the behavior of the maximal Lyapunov exponents of chaotic solutions.Comment: 16 pages, LaTeX (Version 2.09), 10 Postscript-figures included, submitted to Phys. Rev.

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

Bound states in a nonlinear Kronig-Penney model

We study the bound states of a Kronig Penney potential for a nonlinear one-dimensional Schroedinger equation. This potential consists of a large, but not necessarily infinite, number of equidistant delta-function wells. We show that the ground state can be highly degenerate. Under certain conditions furthermore, even the bound state that would normally be the highest can have almost the same energy as the ground state. This holds for simple periodic potentials as well.Comment: TeX file, figures available as postscript files upon reques

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

Simulations of grafted polymers in a good solvent

We present improved simulations of three-dimensional self avoiding walks with one end attached to an impenetrable surface on the simple cubic lattice. This surface can either be a-thermal, having thus only an entropic effect, or attractive. In the latter case we concentrate on the adsorption transition, We find clear evidence for the cross-over exponent to be smaller than 1/2, in contrast to all previous simulations but in agreement with a re-summed field theoretic $\epsilon$-expansion. Since we use the pruned-enriched Rosenbluth method (PERM) which allows very precise estimates of the partition sum itself, we also obtain improved estimates for all entropic critical exponents.Comment: 5 pages with 9 figures included; minor change

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