We have observed unusual nonlinear interactions among "solitary-wave pulses" propagating in nonlinear dispersive media. These phenomena were observed in the numerical solutions of the Korteweg-deVries equation u +uu +du =0. t x xxx This equation can be used to describe the onedimensional, long-time asymptotic behavior of small, but finite amplitude' . shallow-mater waves,~collisionless -plasma magnetohydr odynamic waves, ' and long waves in anharmon-ic crystals.~Furthermore, the interaction and "focusing" in space-time of the solitarywave pulses allows us to give a phenomenological description (some aspects of which we can already explain analytically) of the near recurrence to the initial state in numerical calculations for a discretized weakly-nonlinear string made by Fermi, Pasta, and Ulam (FPU). '»'Spatially periodic numerical solutions of the Korteweg-deVries equation were obtained with a scheme that conserves momentum and almost conserves energy. For a variety of initial conditions normalized to an amplitude of 1.0 and for small 5', the computational phenomena observed can be described in terms of three time intervals. (I) Initially, the first two terms of Eq. (1) dominate and the classical overtaking phenomenon occurs; that is, u steepens in regions where it has a negative slope. (II) Second, after u has steepened sufficiently, the third term becomes important and serves to prevent the formation of a discontinuity. Instead, oscillations of small wavelength (of order 5) develop on the left of the front. The amplitudes of the oscillations grow and finally each oscillation achieves an almost steady amplitude (which increases linearly from left to right) and has a shape almost identical to that of an individual solitary-wave solution of (1).(III) Finally, each such "solitary-wave pulse" or "soliton" begins to move uniformly at a rate (relative to the background value of u from which the pulse rises) which is linearly proportional to its amplitude. Thus, the solitons spread apart. Because of the periodicity, two or more solitons eventually overlap spatially and interact nonlinear ly. Shortly after the interaction, they reappear virtually unaffected in size or shape. In other words, solitons "pass through" one another without losing their identity. Here we have a nonlinear physical process in which interacting localized pulses do not scatter irr ever sibly.It is desirable to elaborate the concept of the soliton, for it plays such an important role in explaining the observed phenomena. We seek stationary solutions of (1) in a. frame moving with velocity c. We substitute u =?T(x -ct) into (1) and obtain a third-order nonlinear ordinary differential equation for u. This has periodic solutions representing wave trains, but to explain the concept of a soliton we are interested in a solution which is asymptotically constant at infinity (u =u at x =+~). The

Two like-signed vorticity regions can pair or merge into one vortex. This phenomenon occurs if the original two vortices are sufficiently close together, that is, if the distance between the vorticity centroids is smaller than a certain critical merger distance, which depends on the initial shape of the vortex distributions. Our conclusions are based on an analytical/numerical study, which presents the first quantitative description of the cause and mechanism behind the restricted process of symmetric vortex merger. We use two complementary models to investigate the merger of identical vorticity regions. The first, based on a recently introduced low-order physical-space moment model of the two-dimensional Euler equations, is a Hamiltonian system of ordinary differential equations for the evolution of the centroid position, aspect ratio and orientation of each region. By imposing symmetry this system is made integrable and we obtain a necessary and sufficient condition for merger. This condition involves only the initial conditions and the conserved quantities. The second model is a high-resolution pseudospectral algorithm governing weakly dissipative flow in a box with periodic boundary conditions. When the results obtained by both methods are juxtaposed, we obtain a detailed kinematic insight into the merger process. When the moment model is generalized to include a weak Newtonian viscosity, we find a ‘metastable’ state with a lifetime depending on the dissipation timescale. This state attracts all initial configurations that do not merge on a convective timescale. Eventually, convective merger occurs and the state disappears. Furthermore, spectral simulations show that initial conditions with a centroid separation slightly larger than the critical merger distance initially cause a rapid approach towards this metastable state.

tour dynamics (CD) for inviscid incompressible fluids in two dimensions.We present a contour dynamics algorithm for the Euler equations of fluid dynamics in two dimensions. This is applied to regions ofThe CD method does not use an underlying lattice and piecewise-constant vorticity within finite-area-vortex regions is a generalization of the ''water-bag'' model used to study (FAVRs). Essentially, this reduces the dimensionality by one and plasma dynamics [5,6]. In essence, it amounts to a dynamic we are treating the interaction of closed polygonal contours whose interaction among closed contours enclosing FAVRs. That nodes are advected by the total fluid motion computed self-consisis, we have reduced the dimensionality by one. To obtain tently. A leapfrog centered scheme is used for temporal advancement. Computer simulation results are given for two and four like-this great simplification, we assume that each FAVR has signed interacting FAVRs. In all cases wavelike surface deformations a constant vorticity density of arbitrary magnitude.are observed. If the distance between FAVRs is comparable to their

We consider the evolution of an isolated elliptical vortex in a weakly dissipative fluid. It is shown computationally that a spatially smooth vortex relaxes inviscidly towards axisymmetry on a circulation timescale as the result of filament generation. Heuristically, we derive a simple geometrical formula relating the rate of change of the aspect ratio of a particular vorticity contour to its orientation relative to the streamlines (where the orientation is defined through second-order moments). Computational evidence obtained with diagnostic algorithms validates the formula. By considering streamlines in a corotating frame and applying the new formula, we obtain a detailed kinematic understanding of the vortex's decay to its final state through a primary and a secondary breaking. The circulation transported into the filaments although a small fraction of the total, breaks the symmetry and is the chief cause of axisymmetrization.