In[]:=
​​eq0=I*D[u[
ϕ
n
,τ]]+(1+μ*Abs[
ϕ
n
]^2)*(
ϕ
n+1
*Exp[I*k]+
ϕ
n-1
*Exp[-I*k])-ν*(Abs[
ϕ
n
]^2)*
ϕ
n
;​​eq0/.
ϕ
n
Function[{n,t},
χ
n
(Exp[-I*τ])]​​%/.{
ϕ
n
Function[ξ,Sqrt[μ]*
χ
n
]}//Simplify
Out[]=
-ν
2
Abs[Function[{n,t},
χ
n
Exp[-τ]]]
Function[{n,t},
χ
n
Exp[-τ]]+(1+μ
2
Abs[Function[{n,t},
χ
n
Exp[-τ]]]
)
-k

ϕ
-1+n
+
k

ϕ
1+n
+u[Function[{n,t},
χ
n
Exp[-τ]],τ]
Out[]=
-ν
2
Abs[Function[{n,t},
χ
n
Exp[-τ]]]
Function[{n,t},
χ
n
Exp[-τ]]+
-k

(1+μ
2
Abs[Function[{n,t},
χ
n
Exp[-τ]]]
)
ϕ
-1+n
+
2k

ϕ
1+n
+u[Function[{n,t},
χ
n
Exp[-τ]],τ]