I have only:

Thank you My version of Mathematica is different and does not give that answer

This should work in version 12:

eq0 = D[u[x, t], {t, 2}] - (v - 2)*
    u[x, t] - (v + v1/\[Mu])*(-u[x, t]^3 + u[x, t]^5);
eq0 /. u -> Function[{x, t}, U[Sqrt[k] (x - c t)]]
% /. Sqrt[k] (x - c t) -> \[Xi]
% /. {U -> Function[\[Xi], Sqrt[W[\[Xi]]]]} // Simplify
Collect[%*4 \[Mu] W[\[Xi]]^(3/2)/(4 c^2 \[Mu] W[\[Xi]]),
 {W'[\[Xi]], W''[\[Xi]]}]

Much obliged.

Here is my interpretation:

eq0 = I*D[
    Subscript[\[Phi], 
     n][\[Tau]], \[Tau]] + (1 + \[Mu]*
      Abs[Subscript[\[Phi], n][\[Tau]]]^2)*
   (Subscript[\[Phi], n + 1][\[Tau]]*Exp[I*k] + 
     Subscript[\[Phi], n - 1][\[Tau]]*Exp[-I*k]) -
  \[Nu]*(Abs[Subscript[\[Phi], n][\[Tau]]]^2)*
   Subscript[\[Phi], n][\[Tau]]
% /. Subscript[\[Phi], n_] -> 
  Function[\[Tau], Subscript[\[Chi], n] Exp[-I*\[Theta]*\[Tau]]]
FullSimplify[% /. Subscript[\[Chi], n_] :> Subscript[u, n]/Sqrt[\[Mu]],
 Element[\[Theta] | \[Tau] | Subscript[u, n_], Reals] && \[Mu] > 0]
Collect[%*\[Mu]^(
   1/2)/(E^(-I (k + \[Theta] \[Tau])) (1 + (Subscript[u, n])^2)),
 {Subscript[u, n - 1], Subscript[u, n + 1]}]

I don't know how to get rid of the exponentials.

This eliminates some common factors and brings the expression into a form more similar to equation (3).

Your equation (1) depends on k, while equation (3) does not. How is it possible? Are you tacitly assuming that k is an even integer?

It was resolved ,Thank you. I couldn't have done it without you.