@Carl Love this is not about result this is have another trick which i can't see , as you saw the paper how they did that? in all paper they did this but why i can't reach the result? i have a series for get eq(16) but is not applying long wave limit,  so they did something which we can't figure out,  i have to figure out , maybe someone did that before and someone intrested like me , i hope  find  this 

@dharr 
I honestly think the problem itself is not too difficult, but the method for finding the solution is quite challenging. We don’t see it the same way the authors do. In many papers, they plot this line directly, but for me, it’s a big challenge, I can't even figure out how they’re getting it.

I need to finish this work, but this approach to solving the problem is causing me a lot of difficulty. In my next question, you’ll see that they use a different method, if I can understand how to apply that limit and use it to find the line, it could help me a lot. Right now, I'm working on figuring out how to determine this line for all the cases. It seems to be related to velocity in some way.

Please, your help on my next question would mean a lot to me.

@janhardo  my deepseek is like that i can't work with it!

about  Bäcklund transformations, Lax pairs i didn't work with them now, but there is more simple work with bilinear form we don't need package even i can get any bilinear form without using this, but for finding auxiliary i have problem  for my last equation when auxiliary is u=u0+ln(f)_x  try this method maybe you can generate a series 

https://www.mapleprimes.com/questions/239878-How-Collect-A-Fraction-Which-Contain
try use this series i am sure they use this to find that kind of auxiliary 

I didn't save last file , nonsense they do dublicate most of questions 

@janhardo i will explain more on  mail send me one please, have another trick which you don't know and maybe you can find general one with this special Ai you have 

@janhardo i like this topic and really is important to me you can find for all by this method ? also if this method is good so you have tt construct for (1+1)D and (2+1)D and (3+1)D also if possible for (4+1) additionally this is the  impotant part of solving some of equation i have  really they  hard  auxiliary which i can't find it , for example this kind i didn't found them even one of them untill now, and please upload the file 

@dharr they say maple can't do that but i told them there is nothing which maple can't but I accepted know by your comment thank you dear Dr.david

@mmcdara  after i finish my work about soliton then i will come to this kind of fluid and i will try use maple for sure, i am sure maple can do that work  too, ofcurse you and Dr.david and moderator and so on are best helper in here 

@mmcdara  they use matlab because i think they have a lot of iteration and they use special pc 

this is so good topic , i liked, but this is not apply for all function i try to solve them more but this method not good for finding all and maybe i am wrong about that i will update some pde here if you can finding auxiliary function u=ln(f)_x it will be soo good , if you need any help you can count on me for finish this invistigate 

U.mw

@mmcdara  just one another step if you do for me i am be thankfull so much, your explanation are so good but in fact i have to do by myself with a lot example until learn better than you  but i don't have enoght time like a beginner  start this topic ofcurse i will back to this topic but when i have time without stress, at  PhD course  time is the important, anyway 

you did one time before i have that code  and i saved about (M+1/G') and you make a good system of coeficent  like list all of phi coeficent  can you make a system of this phi but first replace the S,S1 then substitute and solve system for finding parameter ? like in picture i need that phi but first i have to replace S and S1 ,

something like that 

@mmcdara i think you know how do this kind of technique, do you have anything new about this kind of solution for pde new method like that?

also something is missing i want be sure about that in last step you did there is a G(xi) and  diff(G(xi),xi)   which is i think have to replace by taking derivative from S diff(S,xi) then substitute becuase some part of coeficent contain diff(G(xi),xi) and G(xi) after that we can make a sequence of all coificent of W , we can find a parameter like he did? i did some change i hope they are correct

find-parameter_2_mmcdara.mw

@dharr is give me answer but is not true! This is so different I have to generate system for all kind of finding parameter this is one of them if you need I will upload (2+1) but is so different from this, in other problem I use different way of billinear 

@dharr in last code which i solved it have zero, becuase of that i did! i don't know my parameter is strange i think is wrong like that becuase u is number and my function must have exponential and some other term which construct the soliton by it ...

@dharr I copied the equation at my other file because of that I didn't notice that thanks dr