i don't know how replace the function which remove the sign of integral even i know the author replaced by sin(t) and t and combined them like the picture but i don't know why my pdes not give me zero 

h-pde-M.mw

in this pde I have to find R, and already in the paper finded but  in my openion is not correct even when i tested is not give me solution, in some paper like that i finded R variable by finding some condition in pde but i want a coding for arrange that condition for example in pde we have 4 function of t like C(t),B(t),N(t),and E(t) i have to change them for finding R  like exchange C(t)=C(t), B(t)=C(t),N(t)=-C(t),E(t)=E(t) also we can equal them to one too, this is one of the condition how many condition we can have for substitute each on like that and tested for finding R, one of this condition will make help to find that R  like  this paper

but my equation is different is this equation  in below 

i tested the author outcome and not make my pde to zero, also dipersion relation without changing any any function of t is so different and i think he just use the penlevy test variable 6/beta*diff(phi,x) as a tranformation but i think is mistake, so where is problem and how we can find all the condition 

pde-condition.mw

in this function contain another function which contain integral but when i replace is just take some of the function and other function is not acceptable i want ask  there is any way for plotting the function when i substitute all other function, is work for csc,sec,cosh, and for other not work and  infinity sign appear likesome other  like t,sin,cos,tan,tanh,sinh, so on 

 t-plot.mw

i did seperating but but is seems is not correct what is mistake in here?

issue1.mw

in here  for jacobieliptic i got two lambda[0] same as paper did but in second one is different and is true make my ode be zero, but is so different from paper 

test-Dr.D.mw