Propagation of constructed c-solitary waves

In the section about c-scaling, we looked for an scaling between c-solitary waves. The idea is to be able to use a c-solitary wave as reference and construct others from there. In particular, we constructed the first and third solitary wave from the second one and compared the results. This produced relative errors of around 2% for the stress and x-velocity and around 8% for the y-velocity. In this section we investigate the behavior of these nearly-c-solitary waves as they propagate.

Since these waves are not perfect c-solitary waves, we can’t expect they will propagate in c-media without change in shape. However, we expect they throw away few energy to become perfect c-solitary waves and then propagate without change in shape.

Therefore, what I am mostly concerned about is that the amount of energy that is thrown away is (qualitatively) small and that after this process they behave as c-solitary waves.

See the repo under ‘nwhm/c-dispersion/constructed_c-solitary_waves/plots/sw1’ for a simulation of the stress. In those figures I track down the wave by setting xlim([xm-10 xm+10]) where xm is the location of the maximum point.

It is clear the wave throws away few energy and, afterwards, it propagates without change in shape. We can also measure the speed of this constructed c-solitary wave and see how much it differs from the speed of the actual c-solitary wave. The speed of the constructed c-solitary wave is 1.4056, the speed of the original c-solitary wave is 1.4075 so the relative error is 0.1346%.

I believe these results are much better than what I expected and I can proceed to study the stability of large amplitude c-solitary waves; i.e., see at which speed they become into shocks. I will wait to see if we can get a better scaling though.