Movies of Multi-dimensional Compactons

The 2006 Los Alamos National Laboratory Report LA-UR-06-0239, On Multi-dimensional Compactons by Philip Rosenau, James M. Hyman, and Martin Staley, contains a short mathematical description of the multi-dimensional compactions of the Kdv-compacton equations. The Figures in this paper are frames from movies of the full simulation. To see the movies, click on the movie links below each figure.

Movie 1 (corresponding to Fig. 1 in the paper) is a color density plot of the position of C_2(3; 1 + 2) compactons support evolving out of an elongated initial support. Their corresponding profiles are displayed in Movie 2 (Fig. 2). Note that since m = a+b, the respective compactons have the same support.

Movie 3a and Movie 3b (Fig. 3) shows the collision of three C_2(2; 0 + 2) compactons with speeds = 2, 1.5, and 1. As in 1-D, the interaction is very clean and the compactons seem to emerge intact leaving in the perimeter of the domain of interaction a small ripple.

Movie 4a and Movie 4b (Fig. 4) shows the collision of two C_2(2; 1 + 1) compactons. The loss of mass of the smaller compacton after the collision and that this ripple is more pronounced than in the C_2(2; 0 + 2) equation.

Movie 5a and Movie 5b (Fig. 5) shows two colliding C_2(3; 0 + 3) compactons. As in the C_2(2; 0 + 2) case, see Fig.(3), the two compactons re-emerge intact after the collision without a measurable loss of mass.

The movies for Figure 6 show the evolution of an elongated initial pulse for C_2(2; 0+3), Movie 6_1a and Movie 6_1b (Fig. 6 lower panel), and C2(2; 1+2) Movie 6_2a and Movie 6_2b (Fig. 6 middle panel). In both cases n = 3, but since b = 3 and b = 2, respectively (see Eq.(8)), in the first case the evolution is faster and the emerging compactons are bigger.

Movie 7a and Movie 7b (Fig. 7) for the C_2(2; 0+3) equation shows the decomposition of initial vertical pulse into a compacton and a pair of traveling blobs which at a later time coalesce into a second compacton. Unlike Fig.(1) where the initial support stays put while emitting compactons, here the perturbed domain as a whole is in motion.

Movie 8a (Fig. 8) shows the collision of two 3-dimensional C_3(2; 0+2)compactons. Movie 8b shows the collision of three 3-dimensional C_3(2; 0+2)compactons.

Movie 9 (Fig. 9) shows the emergence of 3-dimensional C_3(2; 0 + 3) compactons out of an initial 3-D ball which breaks into a sequence of toruses, each which later collapses into a compacton.

The number of the emerging compactons and their spatial location depends on the geometry of initial conditions and their span. In Fig.(1) the initial perturbation stays put as it emits compactons. Such scenario is typical of 1 D patterns. On the other hand, emission of 4 compact pulses which are not compactons but converge into compactons as they travel; see Figs.(6) and (9), does not seem to occur in 1 D. Also, as seen in Fig.(7), in higher dimensions we observe patterns wherein the initial pulse propagates as a whole while evolving and emitting compactons. FIG. 8: Collision of two 3-dimensional C_3(2; 0+2)compactons. The early stage of the interaction of their supports at t = 5 and immediately thereafter, at t = 15.

Figs.(3-5) and (8) present the results of hard collisions which occur when the centers of colliding compactons are aligned. Soft collisions which seem more like a skirmish happen when the center of the faster compacton is off center from its prey. When the centers of softly colliding compactons are suføciently close, the fast and the slow compactons exchange their positions. As a rule soft collisions seem to be less elastic then their hard counterparts. Some hard interactions appear to be much closer than others to being elastic with the cleanest interaction undoubtedly reserved to m = n = 2; 3 and a = 0 compactons: C_N(2; 0 + 2) and C_N(3; 0 + 3).

Interaction between compactons: As in 1- D, we observe compactons emerging intact in both 2 and 3 D cases, Figs. (3) and (8). We also studied C_2(3; 1+3) and C_2(4; 1+3), shown in the movies below. Here even though m = n, a = 1 cases are less elastic than in the m = n, a = 0 cases. Thus, while the larger compactons hardly loose any mass, there is a small but noticeable loss of mass in smaller compactons. Repeated interactions enhance this effect. The interaction of 'parabolic compactons' , m = 2 and n = 3: C_2(2; 0+3) and C2(2; 0+3) is far less robust. In collisions the smaller compactons re-emerge greatly diminished, occasionally accompanied with chunks of splitting of pieces of mass.