Pattern Formation And Solitons

We obtain analytical expressions for an effective potential of interaction between two- and three-dimensional (2D and 3D) solitons (including the case of 2D vortex solitons) belonging to two different modes which are incoherently coupled by cross-phase modulation. The derivation is based on calculation of the interaction term in the full Hamiltonian of the system. An essential peculiarity is that, in the 3D case, as well as in the case of 2D solitons with unequal masses, the main contribution to the interaction potential originates from a vicinity of one or both solitons, similarly to what was recently found in the 2D and 3D single-mode systems, while in the case of identical 2D solitons, the dominating area covers all the space between the solitons. Unlike the single-mode systems,_stable_ bound states of mutually orbiting solitons are shown to be possible in the bimodal system.

A general method to find an effective potential of interaction between far separated 2D and 3D solitons is elaborated, including the case of 2D vortex solitons. The method is based on explicit calculation of the overlapping term in the full Hamiltonian of the system (_without_ assuming that the ``tail'' of each soliton is not affected by its interaction with the other soliton, and, in fact,_without_ knowing the exact form of the solution for an isolated soliton - the latter problem is circumvented by reducing a bulk integral to a surface one). The result is obtained in an explicit form that does not contain an artificially introduced radius of the overlapping region. The potential applies to spatial and spatiotemporal solitons in nonlinear optics, where it may help to solve various dynamical problems: collisions, formation of bound states (BS's), etc. In particular, an orbiting BS of two solitons is always unstable. In the presence of weak dissipation and gain, the effective potential can also be derived, giving rise to bound states similar to those recently studied in 1D models.

We show that the linear-stability analysis of the birth of Faraday waves on the surface of a fluid is simplified considerably when the fluid container is driven by a triangle waveform rather than by a sine wave. The calculation is simple enough to use in an undergraduate course on fluid dynamics or nonlinear dynamics. It is also an attractive starting point for a nonlinear analysis.

A new type of transverse instability in dispersively nonlinear optical cavities, called the optical whistle, is discussed. This instability occurs in the mean-field, soliton forming limit when the cavity is driven with a finite-width Gaussian beam, and gives rise to oscillation, period doubling, and chaos. It is also seen that bistability is strongly affected due to the oscillation within the upper transmission branch. The phenomenon is interpreted as a mode-mismatch in the soliton formation process and is believed to have broad applicability.

We consider the bifurcation problem u ′′ +λu=N(u) with two point boundary conditions where N(u) is a general nonlinear term which may also depend on the eigenvalue λ . We give a variational characterization of the bifurcating branch λ as a function of the amplitude of the solution. As an application we show how it can be used to obtain simple approximate closed formulae for the period of large amplitude oscillations.

We show that the minimal speed for the existence of monotonic fronts of the equation u t =( u m ) xx +f(u) with f(0)=f(1)=0 , m>1 and f>0 in (0,1) derives from a variational principle. The variational principle allows to calculate, in principle, the exact speed for arbitrary f . The case m=1 when f ′ (0)=0 is included as an extension of the results.

A discrete phi^4 system is proposed which preserves the topological lower bound on the kink energy. Existence of static kink solutions saturating this lower bound and occupying any position relative to the lattice is proved. Consequently, kinks of the model experience no Peierls-Nabarro barrier, and can move freely through the lattice without being pinned. Numerical simulations reveal that kink dynamics in this system is significantly less dissipative than that of the conventional discrete phi^4 system, so that even on extremely coarse lattices the kink behaves much like its continuum counterpart. It is argued, therefore, that this is a natural discretization for the purpose of numerically studying soliton dynamics in the continuum phi^4 model.

We consider domain walls (DW's) between single-mode and bimodal states that occur in coupled nonlinear diffusion (NLD), real Ginzburg-Landau (RGL), and complex Ginzburg-Landau (CGL) equations with a spatially dependent coupling coefficient. Group-velocity terms are added to the NLD and RGL equations, which breaks the variational structure of these models. In the simplest case of two coupled NLD equations, we reduce the description of stationary configurations to a single second-order ordinary differential equation. We demonstrate analytically that a necessary condition for existence of a stationary DW is that the group-velocity must be below a certain threshold value. Above this threshold, dynamical behavior sets in, which we consider in detail. In the CGL equations, the DW may generate spatio-temporal chaos, depending on the nonlinear dispersion.

How can a system in a macroscopically stable state explore energetically more favorable states, which are far away from the current equilibrium state? Based on continuum mechanical considerations we derive a Boussinesq-type equation which models the dynamics of martensitic phase transformations. The solutions of the system, which we refer to as microkinetically regularized wave equation exhibit strong oscillations after short times, thermalization can be confirmed. That means that macroscopic fluctuations of the solutions decay at the benefit of microscopic fluctuations. First analytical and numerical results on the propagation of phase boundaries and thermalization effects are presented. Despite the fact that model is conservative, it exhibits the hysteretic behavior. Such a behavior is usually interpreted in macroscopic models in terms of dissipative threshold which the driving force has to overcome to ensure that the phase transformation proceeds. The threshold value depends on the amount of the transformed phase as it is observed in known experiments. Secondly we investigate the dynamics of oscillatory solutions. We present a formalism based on Young measures, which allows us to describe the effective dynamics of rapidly fluctuating solutions. The new method enables us to derive a numerical scheme for oscillatory solutions based on particle methods.

We present the first quantitative verification of an amplitude description for systems with (nearly) spontaneously broken isotropy, in particular for the recently discovered abnormal-roll states. We also obtain a conclusive picture of the 3d director configuration in a spatial period doubling phenomenon involving disclination loops (CRAZY rolls). The first observation of two Lifshitz frequencies in electroconvection is reported.