Traveling Wave, EMT/MET transition

We are thinking of EMT/MET transition.

With certain modelling, 

rosa.uniroma1.it

https://hal.science/hal-02520861v2/document

 

When we want traveling wave solution with asymptotic states, we take some function depenedent on (x - ct), where x is the position, and t is time, and c is the speed of propageting front.

 

But, numerically it is difficult to calculate the speed. so we approximate by comparing the previous time function and current time function.

 

Comment. Now I am interested in the rigorous proof of the existence of traveling wave.