Hello!
Thanks for your idea, which helped me out (although a and b might be complex in my case as well).
Now I get algebraic implicit solutions with RootOf(...) - terms. I got some of them away with the allvalues call, but still not everyone was substituted by this call.
I have the work sheet uploaded.
Greetings,
yadaddy.

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Hello!
Thank you!
The result was nice, was it? :)
Many sqrt(1-b^2) terms in there, which I have to replace.
I think the result will get itself easier, if Maple uses a^2+b^2=1 during calculations.
To the real/complex problem:
In complex numbers the root is well-defined on a 2:1 (universal) covering over IC.
The variety of a^2+b^2=1 with a,b in IR or a,b in IC are different:
a,b in IR: circle in IR^2
a,b in IC: surface in IC^2.
So I don't know, if Maple calculates all possible solutions using a=zeta*sqrt(1-b^2), zeta^2=1.
I hope there will be a possibility to use the assumption anyway.
Greetings,
yadaddy.

Hello to each helper!
I think the solution makes geometrical meaning, and I am very thankful for help!
Best regards,
yadaddy.

I found two wrong signs, hopefully now it is correct:
PDE := {
diff(alpha[1](u, v), u) = -I*exp(I*u)*beta[1](u, v),
diff(alpha[1](u, v), v) = -I*exp(-I*v)*beta[2](u, v),
diff(alpha[2](u, v), u) = I*exp(-I*u)*beta[2](u, v),
diff(alpha[2](u, v), v) = -I*exp(I*v)*beta[1](u, v),
diff(beta[1](u, v), u) = -1/4*I*exp(-I*v)*alpha[2](u, v)-1/2*I*beta[1](u, v)-1/2*I*exp(-I*(v+u))*beta[2](u, v),
diff(beta[1](u, v), v) = -1/4*I*exp(-I*u)*alpha[1](u, v)-1/2*I*beta[1](u, v)+1/2*I*exp(-I*(v+u))*beta[2](u, v),
diff(beta[2](u, v), u) = -1/4*I*exp(I*v)*alpha[1](u, v)-1/2*I*exp(I*(v+u))*beta[1](u, v)+1/2*I*beta[2](u, v),
diff(beta[2](u, v), v) = 1/4*I*exp(I*u)*alpha[2](u, v)+1/2*I*exp(I*(v+u))*beta[1](u, v)+1/2*I*beta[2](u, v)};
The funny thing is, that in my Maple output the sign in the 2nd beta-equation
diff(beta[1](u, v), v) = -1/4*I*exp(-I*u)*alpha[1](u, v)-1/2*I*beta[1](u, v)+1/2*I*exp(-I*(v+u))*beta[2](u, v),
at
-1/2*I*beta[1](u, v)
is different then the input I gave to Maple. Why that? I have opened a new sheet, too, and entered it again, but the sign is only in the output the wrong (+ instead of -). But he still continues to calculate with the minus-sign, because I entered:
> DEtools[rifsimp](expand(PDE));
and got the answer:
table([Solved=[(∂)/(∂u) alpha[1](u,v)=-I (e)^((I u)) beta[1](u,v),(∂)/(∂u) alpha[2](u,v)=(I beta[2](u,v))/((e)^((I u))),(∂)/(∂u) beta[1](u,v)=-(1/4 I ((e)^((I u)) alpha[2](u,v)+2 (e)^((I v)) (e)^((I u)) beta[1](u,v)+2 beta[2](u,v)))/((e)^((I v)) (e)^((I u))),(∂)/(∂u) beta[2](u,v)=-1/4 I ((e)^((I v)) alpha[1](u,v)+2 (e)^((I v)) (e)^((I u)) beta[1](u,v)-2 beta[2](u,v)),(∂)/(∂v) alpha[1](u,v)=-(I beta[2](u,v))/((e)^((I v))),(∂)/(∂v) alpha[2](u,v)=-I (e)^((I v)) beta[1](u,v),(∂)/(∂v) beta[1](u,v)=1/4 (-I alpha[1](u,v) (e)^((I v))-2 I (e)^((I v)) (e)^((I u)) beta[1](u,v)+2 I beta[2](u,v))/((e)^((I v)) (e)^((I u))),(∂)/(∂v) beta[2](u,v)=1/4 I ((e)^((I u)) alpha[2](u,v)+2 (e)^((I v)) (e)^((I u)) beta[1](u,v)+2 beta[2](u,v))]])
Seems like to be my input PDE and not the output Maple gave me by pressing ENTER after the input written.
Is the zero-solution still unique or is there another possible solution solvable by Maple?
Thanks,
yadaddy.

Hi!
Where do you know from, that there are no more solutions?
Solving this PDE arises from a geometrical construction. Theory says that there is a solution such that beta is not zero, because you have later divide throught it (looking at beta as a quaternion).
Greetings,
yadaddy.

Hi!
Sorry for being so late now.
Here's the code:
>
> with(PDETools);
> PDE := {diff(alpha[1](u, v), u) = -I*exp(I*u)*beta[1](u, v),
diff(alpha[1](u, v), v) = -I*exp(-I*v)*beta[2](u, v),
diff(alpha[2](u, v), u) = I*exp(-I*u)*beta[2](u, v),
diff(beta[2](u, v), v) = 1/4*I*exp(I*u)*alpha[2](u, v)+1/2*I*exp(I*(v+u))*beta[1](u, v)+1/2*I*(beta[2])(u, v),
diff(alpha[2](u, v), v) = -I*exp(I*v)*beta[1](u, v),
diff(beta[1](u, v), u) = -1/4*I*exp(-I*v)*alpha[2](u, v)-1/2*I*beta[1](u, v)-1/2*I*exp(-I*(v+u))*beta[2](u, v),
diff(beta[1](u, v), v) = -1/4*I*exp(-I*u)*alpha[1](u, v)+1/2*I*beta[1](u, v)+1/2*I*exp(-I*(v+u))*beta[2](u, v),
diff(beta[2](u, v), u) = -1/4*I*exp(I*v)*alpha[1](u, v)-1/2*I*exp(I*(v+u))*beta[1](u, v)-1/2*I*beta[2](u, v)}
Got the problem solved: Maple added "dots" instead of multiplication points.
But now I get only one solution, the trivial one: alpha=0, beta=0.
Is there another way to get a nontrivial solution? I know, there exist nontrivial soltutions.
Thanx,
yadaddy.