@dharr  your idea about hundling thus equation are emazing i liked a lot, as i mention before is a little bit long for explanation but i will try, the procces of solving pdes is like that i have transform the pde for two part real and imaginary then i take real part of pdes as my ode and i will use a methods for solving the ode then this solution of my ode will be solution of my pdes automatically becuase of that i use thus analyticall methods and there is a lot of them which i can't do it yet and becuase construct paper i don't have much time, there is some other way for solvng pdes which i don't have idea but the visualization is great i don't know where to start , like lax pair, lie symmetry  and darboux transformation thus are my future work and keep continuos on analytical methods  when something new come up i will try to solve, and really sorry for that if the questions are a little bit similar, in my mind still i am looking for long wave limit too, thank you a lot and any time have any question i will response and is my honor 

@janhardo don't have link yet

@dharr the whole idea is a little bit long becuase of that i didn't mention all, just the place i stuck, @sand15  my questions are different a lot each time i stuck in a place and i figure out i need your help , and i am really true about that without Dr.David and your help i was not success at all, beside this i am work on unpublished work now becuase of that i need to figure out and construct the idea and i am not sure i am stuck at coding a lot but in idea of undrestanding math i don't have issue at all,  i will upload the picture of one of my article but not published yet and i need to find the solution of all of them and each time use a different ode for getting my solution but the solution are different as clear in picture,

about the parameter at least one of the (F) function must not be zero, and the parameter in orginal equation must not be zero , in here again  i used your code for sech&csch is work but when i want to used for sinh and cosh is not worked and there is one of derivative remain  

Dr.D-thanks.mw

@acer  This version works and is also faster. However, I have another problem the code is too long. I wish there were a much shorter version so I could easily replace my function, and the plotting would be even better.

I also used a 3D plot for that function, but the result is not very visually appealing. If possible, I would like to generate a better-designed 3D plot, as it would be very helpful for my future article. Right now, I am using the following approach.

@sand15  when i run it it take a lot of time visualization is perfect more than matlab even, How about the 3D we can have something like that , but be a little fast work too is will be better

@janhardo  i did this but result is different

@janhardo  i need just one step no more

@janhardo  i need evaluate |u|^2 but by code what code we have to use for get the same result?

@sand15 something wrong in my changing pdes which is related to complex number and i don't have idea about that kind of substitution which work automatically 

@dharr  i try another method but you did the same as paper did , i am looking for a methods for finding this kind of pdes i will try more and more  thanks a lot 

@janhardo i need solving not writing 

@dharr  i know is confusing becuase this topic  have a lot kind of solution of pde, Soliton=5 type, lump=3 type, breather=2 type, interaction between them also have at least 8 type each on of them the solution is change, but i am sure if for the first two one is correct and without doing substitution is give me zero it  will be  done , if you watch carfully you will see after the first solution make pde be zero in (pdetest(eq19, pde)) then when N=2 start we look for second solution  if that solution make our pde be zero then we say that our substitution is true ,  but how find that condition which make our pdes be zero i am sure is related with that part becuase if you want be sure set all of the function of (t) be one, you will see  the second solution is satisfy pdes

@dharr i know where is problem but i don't know how find the correct one watch this picture

i did this change (red part) this red part is not true is not work if do this change it work just for first pde for other is not true, but i know there is a condition with one  exchange (we look for that true exchange) and substitute it can be any exchanging it will work for all so how i find that i will take another shot in below mw file

another-c.mw

@dharr  i solved one another before he didn't make any problem  and without this issue it  give me zero, maybe i am in mistake i trust your way about the last pde which make it to zero by substituting and really it work but know about this which not give it to me zero i am in dubt about that i think  without replacing parameter and function must give me zero, so in here i will sure which i did a mistake and replacing not work is my openion 

in my pde  i did a substitution  if you watch  which is  alpha(t)=beta(t) i did that by myself becuase for finding R i need to do something like that, if i don't do this substitution is not give me zero in first pde but also this substitution must be true for second time, if not work for second so we figure out we did mistake at there at that substitution, i don't know how act with that substitution do you have any idea  about that at least one function of (t) remain in pde  which worked for both pde, there is a lot of pde which if i can find this condition  for them i will be super work

really need that changing of function also we can change them to a constant but at least one of them remain and make our two step pde be zero i am sure if untill second one be zero it will work for other type too, i hope find this i will mark in red color 

C-find.mw

@dharr in here why when i test without substittue parameter is not give me zero but when i do substitute is give me zero?

really is problem how this is possible?

problem.mw