Dromions

Dromions - The movie

This page contains the material previously distributed as a VHS video "Dromions- The Movie" by J. Hietarinta, University of Turku, Turku, Finland and J. Ruokolainen, Centre for Scientific Computing, Espoo, Finland.

The animations were made from analytical formulae computed using REDUCE computer algebra program, the results were converted into C-programs and the data visualized using "funcs" on a SGI workstation.

In this video we show certain types of solitons. In general we may define solitons as localized "lumps" of matter that are conserved and scatter elastically. Such solitons are solutions of certain nonlinear partial differential equations that have many nice properties. Solitons have been observed in nature in various sizes, from very long ocean waves to bondings in acetylene, and in various substances, from magma to optical fiber.

The nice mathematical properties of solitons equations allows one to obtain solutions to them, even though the equation is nonlinear. Of the possible solution methods we use here Hirota's bilinear method. It is the most efficient one for obtaining multisoliton solutions.

	Typically soliton equations have one space and one time variable, as in the nonlinear Schrodinger equation, a two-soliton solution is displayed in adjoining animation.
	In higher dimensions the solitons appear as plane waves. If we want to extend the nonlinear Schrodinger equation to 2+1 dimensions in the naive way given here, we will get only parallel plane waves.
	However, the Davey-Stewartson equation can have solutions with intersecting plane waves. Note that we have introduced an auxiliary function w. In the animation we show the u-function for a two-soliton solution consisting of two intersecting plane waves.

In 1988 Boiti et al found [Phys. Lett. A 132, 432 (1988)] that the Davey-Stewartson equation can have solutions localized in two dimensions.
These results were generalized by Fokas and Santini [Phys. Rev. Lett. 63, 1329 (1989)] using ISM.
Hietarinta and Hirota [Phys. Lett. A 145, 237 (1990)] have obtained still more general solutions using "double-Wronskians" within the bilinear method. These solutions are illustrated here.
The Davey-Stewartson equation given before is a coupled system of two functions u and w.
- u is physical and localized
- w shows the underlying plane waves
The plane waves w are kind of tracks ("dromos" in Greek).
At their intersections we have the localized lumps of u, the dromions.
Below we will show several multidromion solutions. We will always show both the perspective view of u-function and the birds-eye view of both the u- and w-functions.

		Dromion scattering 2->2. This is the type of solution first obtained by Boiti et al. The left-hand side picture is a perpective view of the u-field, on the right we have a top view of u and v. Note that when the w-plane waves cross without interaction (no phase shift) we have no contribution to the u-function. The displayed matrices B, H, and K characterize the solution as discussed in Hietarinta and Hirota.
		Dromion scattering 4->4. The left-hand side picture is a perpective view of the u-field, on the right we have a top view of u and v.The dromions are of equal size and scatter without change in amplitude. This is an example of the solutions obtained by Fokas and Santini, the final amplitudes are determined from the initial ones. Note that in this case the parameter matrices for H and K are nondiagonal.
		Dromion scattering 4->3. The left-hand side picture is a perpective view of the u-field, on the right we have a top view of u and v. This animation shows how a four-dromion initial state can change into a three-dromion one. The initial amplitudes are the same as before, but for there are additional degrees of freedom so that one can construct solutions with the same initial amplitudes as before but different final ones. Note that now the B matrix has an imaginary offdiagonal entry.

Parameter dependence. The animation (left to right) show parameter dependence at the initial, middle and final time, respectively.

One can also construct dromion solutions for other equations. [Hietarinta, Phys. Lett. A 149, 113 (1990)] The essential ingredients are:
1) Definition of the physical function so that there can be plane waves whose physical contribution vanishes ("ghosts").
2) Dromions are made of intersecting ghost solitons.
Example: Dromions for the KP-equation Bilinear form: (D_x^4 + 3D_y^2 - 4D_xD_t)F.F = 0.
Let us define the physical function by u: = (log F)_{yy} - (log F)_{xy} (standard choice is w = (log F)_{xx})
Plane waves parallel to directions (1,0) or (1,1) are ghosts when u is the physical function.

KP-dromions: The left-hand side picture is a perpective view of the u-field, on the right we have a top view of u and w. Note that now we have no freedom to choose the amplitude, it will be determined by the velocity of the underlying plane waves. This means in particular that the amplitudes are conserved during interaction.

Last update: April 11, 2001. - -