Constant solutions to nonlinear Klein-Gordon, NKG, equations are easy to identify.
Not only is 
a solution to all three model equations, but any root
of the function 
will also satisfy eq:NLKG. For example,
the phi-four equation has three constant solutions, 
and 
,
while the sine-Gordon equation has an infinite number of solutions, 
.
Although these solutions are trivial and unintersting in and of themselves,
they often represent stable states of the system. Any soliton would likely approach
such a state as 
. If a system has multiple stable
states, then it is easy to imagine a solution that takes the system from one
stable state to another; such a solution is called a kink if 
increases as x increases and an anti-kink if 
decreases.
Unfortunately, a kink (or anti-kink) is topologically impossible for
our system since all solutions must match the boundary conditions.
Kink/anti-kink pairs are possible and will become the basis for some interesting simulations.[1]
Figure: Space-Time contour of a low energy kink-antikink collision in the phi-Four equation.
Here are a few hints on how to use some of the program's features to complete the following exercises. Try to construct your solutions one piece at a time. You can then edit the two parameters in the formula that determine the initial position and velocity of the soliton. The parser input screen allows you to add new functions to the current wavefunction. It is also possible to merge two WAVE data files together using [Files]|[Merge]
. All existing settings are preserved; only the wavefunction is changed. This will be particularly useful for multi-soliton collisions. Finally, be sure to set the medium to 
since all parameters in the exercises have been selected to give good results for this setting.
using [Parameters]|[Space Parameters]
and increase
the number of grid points to 512. Set the boundary to fixed and select the sine-Gordon equation.
. The system will warn you that it has chosen Klein-Gordon rather
than sine-Gordon eigenfunctions for reasons that will be obvious after you complete the
exercises. Run the simulation with a contour analysis graph.
Notice that the system acts pretty much like the a Klein-Gordon eigenfunction. This is
not suprising since 
for low amplitude waves.
where 
and v is the velocity of the kink.
as before. Use
[Init]|[User Defined Function]
to set the initial state to be a kink,
eq:SGKink, with 
.
It is important that you do not set t
equal to zero. The program will evaluate the function at t=0 to determine yVec
and again
at 
to determine yPrevVec
. Does the kink propagate as expected? Measure the
velocity of the kink. Notice that the boundary introduces a discontinuity.
It is convenient to type each 
function into a separate text field since the two text fields on the parser input screen are concatenated. Run
the simulation with fixed boundaries.
to keep from having to rescale the screen.
Notice how the anti-kinks retain their shape as they pass through each other, even though the system is both dispersive and nonlinear.
. Will such a state propagate without changing shape?
Select the phi-four equation and create a grid with 512 points and periodic boundaries at 
as before.
Place a detector at x=0 and select a detector analysis plot along with a contour analysis plot.
Note: The 
term in the phi four simulation makes the simulation susceptible to numerical overflow-- and a
possible program crash. The chances of this happening are minimized if you select a large number of
grid points.
and an anti-kink
at x=+10 with 
. Make sure that the function you input begins and ends in the
stable x=-1 state and that the kink carries the wavefunction into the stable x=+1 state.
The boundaries should, of course, be periodic so that the ends do not add an unstable x=0 solution.
Run the simulation and observe the soliton-like behavior--- at least for the first
collision.
and an anti-kink
at x=+10 with 
. Run the simulation and describe the motion. The
long-lived bound state is called a breather/bion. Notice the small
traveling waves. They represent radiation leaving the interaction region.
and an anti-kink
at x=+10 with 
and run the simulation. Notice the long collision time.
There are a number of such resonances between 
and 
.
rather than 
.
A kink solution exists and is given by
where 
and 
. You should use the [F3]
hotkey to toggle to a derivative plot
of the wavefunction before running this simulation in order to highlight properties of the soliton.
and anti-kink at x=+10 with 
. An easy way
to do this is to create a kink with the parser and
exit the parser screen. You can then re-enter the parser, create an anti-kink, and add it to the existing kink using the add to existing wave option. Another way would be to create kink and anti-kink files and use [Files]|[Merge]
to combine these functions. Run the program. Are the kinks true solitons? Do they pass through each other or do they interact?
to 5. What happens now? The solutions are no longer solitons but they still propagate a disturbance.
