Pavel M. Lushnikov

Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact.

A connection is established between discrete stochastic model describing microscopic motion of fluctuating cells, and macroscopic equations describing dynamics of cellular density. Cells move towards chemical gradient (process called chemotaxis) with their shapes randomly fluctuating. Nonlinear diffusion equation is derived from microscopic dynamics in dimensions one and two using excluded volume approach. Nonlinear diffusion coefficient depends on cellular volume fraction and it is demonstrated to prevent collapse of cellular density. A very good agreement is shown between Monte Carlo simulations of the microscopic cellular Potts model and numerical solutions of the macroscopic equations for relatively large cellular volume fractions. Combination of microscopic and macroscopic models were used to simulate growth of structures similar to early vascular networks.

Continuous macroscopic limit of a discrete stochastic model for interaction of living cells

We derive a continuous limit of a two-dimensional stochastic cellular Potts model (CPM) describing cells moving in a medium and reacting to each other through direct contact, cell-cell adhesion, and long-range chemotaxis. All coefficients of the general macroscopic model in the form of a Fokker-Planck equation describing evolution of the cell probability density function are derived from parameters of the CPM. A very good agreement is demonstrated between CPM Monte Carlo simulations and a numerical solution of the macroscopic model. It is also shown that, in the absence of contact cell-cell interactions, the obtained model reduces to the classical macroscopic Keller-Segel model. A general multiscale approach is demonstrated by simulating spongy bone formation, suggesting that self-organizing physical mechanisms can account for this developmental process.

Multiscale dynamics of biological cells with chemotactic interactions: from a discrete stochastic model to a continuous description.

The cellular Potts model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. We derive a continuous limit of a discrete one-dimensional CPM with the chemotactic interactions between cells in the form of a Fokker-Planck equation for the evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. In particular, all coefficients of the Keller-Segel model are obtained from parameters of the CPM. Theoretical results are verified numerically by comparing Monte Carlo simulations for the CPM with numerics for the Keller-Segel model.

Nonlinearity management in a dispersion-managed system

We propose using a nonlinear phase-shift interferometric converter (NPSIC), a new device, for lumped compensation for nonlinearity in optical fibers. The NPSIC is a nonlinear analog of the Mach-Zehnder interferometer and provides a way to control the sign of the nonlinear phase shift. We investigate a potential use of the NPSIC for compensation for nonlinearity to develop a dispersion-managed system that is closer to an ideal linear system. More importantly, the NPSIC can be used to essentially improve single-channel capacity in the nonlinear regime.

Dispersion-managed soliton in a strong dispersion map limit.

A dispersion-managed optical system with stepwise periodic variation of dispersion is studied in a strong dispersion map limit in the framework of the path-averaged Gabitov-Turitsyn equation. The soliton solution is obtained by analytical and numerical iteration of the path-averaged equation. An efficient numerical algorithm for finding a DM soliton shape is developed. The envelope of soliton oscillating tails is found to decay exponentially in time, and the oscillations are described by a quadratic law.

Dispersion-managed soliton in optical fibers with zero average dispersion

A dispersion-managed optical system with stepwise periodic variation of dispersion is studied in the framework of a path-averaged Gabitov-Turitsyn equation. The soliton solution is obtained by means of iterating the path-averaged equation. The dependence of soliton parameters on dispersion map strength is investigated, together with the oscillating tails of the soliton.

Collapse of Bose-Einstein condensates with dipole-dipole interactions

In this paper sufficient analytical criteria are developed both for catastrophic collapse of BEC of a trapped gas of dipolar particles and for long-time condensate existence. Sufficient criteria allow one to predict condensate collapse or, contrarily, its long-time existence for given condensate energy E, number of particles N, initial mean-square width of condensate, and initial kinetic energy of condensate. Analytical criteria are compared with results of the variational approach @8#, where collapse was predicted based on the absence of a local minimum of the ground state of the energy functional provided number of condensate particle exceeds a certain critical value. It is shown here that variational calculation gives a threshold number of particles and condensate energy which are located between parameter regions where analytical criteria predict collapse and long-time condensate existence, respectively. It is proven in this paper that collapse certainly occurs provided energy of the condensate is below a threshold value which is determined by the number of particles and trap parameters. Collapse of the condensate is accompanied by a dramatic contraction of the atomic cloud. Collapse is impossible provided the number of particles and initial kinetic energy of condensate are below the critical values. The time-dependent Gross-Pitaevskii equation ~GPE! for atoms with long-range interactions and for a cylindrical harmonic trap is given by @6#

Twin families of bisolitons in dispersion-managed systems

We calculate bisoliton solutions by using a slowly varying stroboscopic equation. The system is characterized in terms of a single dimensionless parameter. We find two branches of solutions and describe the structure of the tails for the lower-branch solutions.

Oscillating tails of a dispersion-managed soliton

Oscillating tails of a dispersion-managed optical fiber system are studied for a strong dispersion map in the framework of a path-averaged Gabitov–Turitsyn equation. The small parameter of the analytical theory is the inverse time. An exponential decay in time of a soliton tail envelope is consistent with nonlocal nonlinearity of the Gabitov–Turitsyn equation, and the fast oscillations are described by a quadratic law. The preexponential modification factor is the linear function of time for zero average dispersion and a cubic function for nonzero average dispersion.

Non-Gaussian statistics of multiple filamentation

We consider the statistics of light amplitude fluctuations for the propagation of a laser beam subjected to multiple filamentation in an amplified Kerr media, with both linear and nonlinear dissipation. Dissipation arrests the catastrophic collapse of filaments, causing their disintegration into almost linear waves. These waves form a nearly Gaussian random field that seeds new filaments. For small amplitudes the probability density function (PDF) of light amplitude is close to Gaussian, while for large amplitudes the PDF has a long powerlike tail that corresponds to strong non-Gaussian fluctuations, i.e., intermittency of strong optical turbulence. This tail is determined by the universal form of near singular filaments and the PDF for the maximum amplitudes of the filaments.