Hi A,
don't be surprised. The identity that produces Sech as the soution to the KdV equation involves the inversion of the elliptic integral F[ArcSinh.] in order to get an solution x-> sech 0 d/dx JacobiAmplitude[x,1].
But the general JacobiAmplitude is probably ill defined in Mathematica. It has jumps by 4 Pi at the peridodicity point +- 4 K on the real line.
All this mess stems from the fact that the JacobiAmplitude is a complex line integral over JacobiDN that has a lattice of simple poles in the complex domai. The choice of the path of integration around all the logarithmic branch points to choice is the decision of the IT-specialist who in general has as little knowldege about elliptic functions as most mathematicians.
This ambiguity is at the heart of the impossibility to verify solutions to our beloved models solvable nonlinear PDE's KdV and sineGordon.
In[28]:= DSolve[ u'''[t] u[t] - u'[t]*u''[t] + u[t]^3 u'[t] == 0,
u[t], t]
Out[28]= {{u[t] ->
InverseFunction[-((2 I EllipticF[
I ArcSinh[(Sqrt[1/(-C[2] + Sqrt[-C[1] + C[2]^2])] #1)/
Sqrt[2]], (C[2] - Sqrt[-C[1] + C[2]^2])/(
C[2] + Sqrt[-C[1] + C[2]^2])] Sqrt[
1 + #1^2/(2 (-C[2] + Sqrt[-C[1] + C[2]^2]))] Sqrt[
1 - #1^2/(2 (C[2] + Sqrt[-C[1] + C[2]^2]))])/(Sqrt[
1/(-C[2] + Sqrt[-C[1] + C[2]^2])]
Sqrt[-2 C[1] + 2 C[2] #1^2 - #1^4/2])) &][t + C[3]]}, {u[
t] -> InverseFunction[(2 I EllipticF[
I ArcSinh[(Sqrt[1/(-C[2] + Sqrt[-C[1] + C[2]^2])] #1)/Sqrt[
2]], (C[2] - Sqrt[-C[1] + C[2]^2])/(
C[2] + Sqrt[-C[1] + C[2]^2])] Sqrt[
1 + #1^2/(2 (-C[2] + Sqrt[-C[1] + C[2]^2]))] Sqrt[
1 - #1^2/(2 (C[2] + Sqrt[-C[1] + C[2]^2]))])/(Sqrt[
1/(-C[2] + Sqrt[-C[1] + C[2]^2])]
Sqrt[-2 C[1] + 2 C[2] #1^2 - #1^4/2]) &][t + C[3]]}}
In[25]:= MapAll[
FullSimplify[
PowerExpand[
Simplify[# /. {(C[2] - Sqrt[-C[1] + C[2]^2])/(
C[2] + Sqrt[-C[1] + C[2]^2]) ->
4/\[Omega]^2 ((-C[2] + Sqrt[-C[1] + C[2]^2])^(-(1/2)) #1)/
Sqrt[2] :> A #1 } ]] ] &,
DSolve[ u'''[t] u[t] - u'[t]*u''[t] + u[t]^3 u'[t] == 0, u[t], t]]
Out[25]= $Aborted
In[9]:= u'''[t] u[t] - u'[t]*u''[t] +
u[t]^3 u'[t] /. {u -> (Sech[ #/2] &)} // FullSimplify
Out[9]= 0
In[27]:= [PartialD]_x JacobiAmplitude[x, 1]
Out[27]= Sech[x]
Regards Roland