@salim-barzani 
What is the bilinear pde derived from your original pde you posted?
From there to get a tau function and applying the longwave limiet for a lump ( f4) 

@salim-barzani 
I haven't done anything yet with the given PDE to derive a Hiroto bilinear form.
And then, from there, set up the tau function.
Didn't I create a procedure (corrected existing procedure) for that bilinear form to obtain it easily?
Remember that this procedure does not accept an integral expression in the PDE, so the procedure is of no use here for this 2sdk PDE.

I did start with an assumed (known?) tau function to obtain the F4 expression, but the desired form has not yet been achieved without errors, as can be seen in the code, so...

@salim-barzani 
The tau function for F4 has been accepted and is not yet symbolically derived?

I did find that f2, but this solution no longer follows the text of the attached papers, so I'll have to figure it out myself?

@salim-barzani 

I don't remember the theory behind solitons either.

I do know that the top of a certain type of soliton moves along a straight line.

I would have to look at it step by step, first for one trajectory line to see which solitons (contours) can be found.

What determines that the contours are on different lines of motion? In short, it doesn't seem like an easy fix.

How about t in your code expressions ?

You can ask a question in the field at Promt.
Still a bit user-unfriendly for input, because you can't jump to a new line?
Not using the latest version of Chat Gpt yet?
Note: ChatGPT can use uploaded pics with maple code,handy. 

with(NaturalLanguage);
GenerateDocument[interactive]();

The wolf best position is in the centre of goat circle ?
Maybe plotting the goat's position per second ?

I think it would also be informative to see the actual wolf prey boundary curve as an overlay as a point plot over the existing procedure simulation plot.

Within this wolf boundary curve, the prey can be caught, and outside it, it cannot.

@dharr 

Thank you, that confirms that the procedure code is working correctly.
Without animation, I don't really know how long the goat spins around.
There is a time calculation when the goat is caught
There is a starting angle , but the goat can start in two directions>
Try to get his in th e code too.

That manual code is a good approach.

Actually, there are only three possible positions for the wolf: outside the circle, inside the circle, and on the circle.
This can easily be calculated manually, so is there really a need for a procedure?

Note: Now the question arises: which direction would the goat have instinctively chosen?
FastPursuit2(1, 1, 1, 0, 5, 0.5, 5, 0.05, direction = "clockwise");
FastPursuit2(1, 1, 1, 0, 5, 0.5, 5, 0.05, direction = "counterclockwise");

After 5 seconds hunting on the goat by the wolf..

For 5 seconds hunting on the goat by the wolf : choosen clockwise by goat , but  in the end not.

achtervolgings_kromme_wolf_en_geit_mprimes_DEF1_15-9-2025.mw

@sand15 
You must use PMF ?
for PMF (Probability Mass Function) → Voor discrete variabelen

I  looked again to the code ,this is clockwise and  FastPursuit2(1, 1, 1, 0, 5, 0.5, 50, 0.05, direction = "counterclockwise") is also possible 
achtervolgings_kromme_wolf_en_geit_mprimes_DEF_14-9-2025_(2).mw
Strange curve from the wolf to  go to the goat ?

Improved FastPursuit2 procedure

FastPursuit2(5, 1.5, 1.0, 0, 2.0, 0., 1200, 0.05) example: the wolf is running in a almost concentric circle at almost fixed distant (not this plot)

achtervolgings_kromme_wolf_en_geit_mprimes_DEF_14-9-2025_.mw

achtervolgings_kromme_wolf_en_geit_mprimes_14-9-2025_.mw

Making now a FastPursuit2  procedure using dsolve :
sol := dsolve(sys union ics, {xw(t), yw(t)}, numeric, method = rkf45)

@Alfred_F 
 

Starting position and speed are decisive?

The goat continues to move in a circle

The wolf remains focused on the goat

Was it not with a pursuit curve?