@janhardo thanks john, it is something new , so i am looking for somehow  merge all part  together if this is a line How related to time variable t , when we have three lump in one graph with different time  How i can merge this with my code and give me my graph this is so importan ,it seems you know what you do and you know the idea behind this which i don't know it will be great if finish this work 

@dharr  i just looking for get new  idea but nothing come out  , i will updload some of this paper in case you see something new  but in all  of them don't saying anything about this , also john did something strange which make me confuse more  he find slop which you founded last time by dy/dx and used in slop equation  and make U(x,y,a,b) and instead of y i think he used function y=y0+m*(x-xo) instead of each a and b constant and ploted, x0 and y0 are intial, which i think is all of this are too close for reach but i can't know how , but i know is related to this equation  y=y0+m*(x-xo) all of them have this shape john combined them but he can't reach out too in each side something messing i needed this part for solve my puzzle 

the next paper is  which i did this too but can't reach line y again https://www.mapleprimes.com/questions/239905-How-Find-Line-For-Plotting-

lump-graph-john.mw

@janhardo  can you plot three lump last time you did also the color of each lump be different like picture 

@mmcdara  sorry i didn't know

@dharr incrideble, i will try find for other equations i hope no problem come up, dear Dr.David How about plot of this line along with lump , john did a plot for last problem but i can't read the code is a little bit  strange but the result is so clear  i upload him code here in case to get the idea of plotting and recoding , 
and i have a question, How define this 
Eqs 16,17
eqp := {x = xp - 5*(a^2 + b^2)*t, y = 10*a*t + yp};
after substitution must take it from this equation 
eqfcomplex := collect(evalc(eval(eval(eqf, p[2] = conjugate(p[1])), p[1] = a + b*I)), t);
my openion about that i tested and i take something which i don't know is true or not?
if we take a derivative from eqfcomplex by x when y=0 and by y when x=0 , we have to substitute the outcome part in (eqfcomplex) and we get our equation  which is  eq (16), also eq(17) but how substitute?

Dr-D-Line-derivative_add.mw

 line-lump-graph-John.mw

@dharr & @mmcdara  thank for both of you i combine them and the rest i will work on it , if possible work on this i stuck bad at here https://www.mapleprimes.com/questions/239914-How-Find-Line-In-Pde-Which-Can-Cut-Each

Dr.D-macdara-pde.mw

@dharr  i think the nonlinear part is wrong c[1] u[x$3,y] is linear but in seperating setting like non linear ? How?

Dr.D-pde.mw

@mmcdara  Dr.David make this and it answer to almost all of them except some of this question when we substitute we get (R) but in here we have zero, i don't know how find this (R) must be another way , i though maybe pde is wrong but is strange 

@dharr 

https://doi.org/10.1007/s11071-020-05611-9 (copyright material removed by moderator)

@dharr Dear Dr, Dr.macdara   answered this i saved him answer and worked, but i am very match with your code, and detail, and tonight i will delete https://www.mapleprimes.com/questions/239905-How-Find-Line-For-Plotting-  becuase i didn't reach the solution, and they don't let me asked another question like this even different,  i have 2 or 3  another paper to show How they reach this in this topic just this part remain, which really i am count on you to reach the solution , which will be finish just one step remain i hope tonight this step ended , i keep working to solve more paper to reach out ,but i can't see problem.
and he answered like this maybe get new idea
stop-working_mmcdara.mw

@janhardo  we can't see that trick but i try to post a new question and more example in case someone have idea for finding, becuase dublicate maybe i delete this if don't have problem

@janhardo i have enough paper and i have to solved but in all of them i can't find this line i have to show more example today i will try finish my job and collect all information in one package with more than at least three or 4 example to how find this line and velocity.

@Scot Gould i know you don't answer my question and you know them but  my questions are not same they mention in tag dublicate if i online i will remove dublicate but at that time even i don't know they taged dublicated, i just see notification which you answered!  i didn't delete thus which you answer and really it will be so good if you answer  watch even this post in top, it is not dublicated question but they mention is dublicated is not my false and sometime when in 24 hour i didn't get my answer i will delete and again asked same question but with more detail and they say you don't have to do that , and this act is just seeking for get answer in short time with update detail ,even in this post i try to solve another example but needed time for collecting information

@janhardo How find this line? How find eq(11&12)? How Find v[x]&v[y] this is important part, if i find in one case i will do it for other just need this each part seperatly 

@janhardo i delete the question becuase they didn't  let me ask another one with the same topic, yes i think there is a lot people who know about that i don't want go in this convesation , you did and it is correct the graph is exactly what i want but importan thing is this How we reach this result there is a 1000 equation we need find for them at there we don't have solution for watching we need finding, i need a way for finding  each part and structure step by step ,