@dharr 

I’ve been studying this carefully, and in your code (and in the paper) they obtained three specific functions, u2,u3,u5 while the others remained free. I’m trying to deepen my understanding of the Painlevé test and its steps. I’m still not entirely clear why it is useful, or why certain positions are called “resonances” (some of which are zero and some not).

The main reason I’m focusing on the Painlevé test is that I recently came across a new method (with only one published paper on it so far). I’m convinced that if I can fully understand this test, I’ll be able to publish several papers, but at the moment I’m stuck.

I’ll update you with the link soon; if you can’t download the paper, let me know and I can send it by email. I can also share a picture of the equations, I haven’t skipped to the end, but after doing the transformation, some very complicated equations appeared. I don’t fully understand how they derived them, so maybe your version is different.

I know you’re busy, but I just wanted to let you know in case you’re also interested.

specially becuase of this i study penlevy test 

i did transform but i didn't penlevy yet

paper

 F-ode-Fpaper.mw

@dharr is more than perfect and in some place really i have to work on it more specially of thus plot you did is novel idea in a=-1 and a=1 have two intersection but in other place not , i have to check for other pdes too i need some time for that, beside this i add one part in third step which you did for finding u[1] and u[2] in last file but in here you didn't mention it i add that part for finding u[1]-u[6] when we have u[1]=() it mean free, and how you know at j=1 when simplify is zero why in j=4 is not zero at that resonance point?

also did you work on couple system is so importan if we complete that part, i have problem at there a lot and today i didn't have much time to work on it sorry for that i will keep looking 

all-steps-Dr.D.mw

@dharr  it seems in solving schrodingers he use the coupled system like the paper you mentioned, i  didn't work on couple system but i wil search more  and i will try to solve 

@dharr new update for step one

schrodinger-test-1.mw

@dharr this is penleve test again but more complicated with same steps, How we can find r and p of substitution is like before when we try to find (a) in non shcrodinger equation but this time we have two function i saw in a paper how they find but we need find all posiblety  or just high order and nonlinear term for two function which one get u=a[1] and v=a[2] and i think after substitute u=u[0]F(x,t)^p in real part and calculate p and r at there also in imaginary but i think both will be equal, beside this how find u[0] and v[0] is the same of step but have two function i try to substitute but i don't know my code why not work specially for finding p and r u(0) v(0) also j point 

i got the idea of p and r from this paper 

schrodinger-test-1.mw

@dharr I have to work on it more for full understanding the topic and thus finding is just begin I will try find all solution I hope to not stuck... 

again thank a lot....

@dharr  i will test for more pdes and for sure i am go to next topic after i got expert in finding thus but i think some problem will come up it seems becuase i need find all function, i will do the rest for sure if i have a question i will ask again , and is a gate for backlund transformation also there is another way for finding same step for shcrodinger equation which after this i will go apply for that but  undrestanding this is mean undrestanding all the gate to other topic.

i don't know how say thank you without you is imposible to done this you  made my night again 

@dharr in a lot of place  but very small place they talk about this please watch thie yellow place of step3 of this article i think if we do the change of variable like that we can solve it, in a some other place they talk about this replacement

e-kdv-burger-equation.mw

i try to solve but i got error, and we use the series wrongly i think, becuase phi(x,t)* series to all term must have that part

@dharr  i did by old version of code for some pdes the a give me wrong i will use this now for clerify, i can solve byy hand but code is something else,and i am try to figure out this test is good for what i hope  untill then you can figure out the step 3, and i am sure this test give us some pdes solution too 

there is any different between 2024-2025? if i know is make a prolem for your code i will change to 2025

@dharr  i try to fix it but not working, and t is very compact 

@dharr I am kepp searching for more example, but in all paper which I saw they didn't tell what they did just I saw in that paper I mention up, tomorrow I will search more , any idea come in my mind  I will update here ...

@dharr  i don't know how your code is work, you deserve to be called by god of maple...

some code give me error in first step but i replaced by your last file to fixed, it work , and it is emazing you have done the step two i will do test for more of them tommorow, about the step 3 your step is true but i think when you replacing w=W in eqs2 i think all of them not change maybe at there is mistake ,even if you watch the indets you will figure out, and maybe i am wrong but i determine that point , also i change that series you used,  have three function we just have two function which one of them we take for get coiefecient i determined in file, there is something in paper i am not confortable with it  which i am don't know what is mean of that (By substituting Eq. (14) into Eq. (4) and collecting the terms of the same powers of and equating them to zero,) same power what mean of this

GOM-Dr.D-test-1-2.mw

@dharr is so good not just corect, and in end of the work i will compact more...

@dharr if you have time please fix that part step two and three are the same i think i try to fix but i can't see the error ,but you do

@dharr  step one done completly- i will go combine your code i am busy with it a lot and i am in step two i can't figure out  how they j point why is not appear is not hard but  something is there about coding i can't hundle it, 

for your question about nonlinear term

you have to chose  the big power or order or multiply of them together for example if we have something like that in one pde term
pde1=u_xx+u^5+u*u_x+u_x*u_xx...

highorder=u_xx=a-2

nonlinear term=u^5=5*a is highest in pde1

if we done have u^5 term in pde1

pde2=u_xx+u*u_x+u_x*u_xx...

highorder=u_xx=a-2

nonlinear term=u_x*u_xx=a-1+a-2

i need your help for finishing this emazing job and this 3 steps  is just begin, is a gate for two other topic like backlund transformation and lax pair but in begin i have to do this , and if you have any confusing in pdes about nonlinear term mention me , and i did for two pdes in step one by the old code but stucked in the step2, if need anything please mention 

pdes-s-1-2.mw

step1-2.mw

i will go combine your code and keep work on it, thank you so much your job is valuable