## 96 Reputation

16 years, 71 days

## Hello! Thanks for your idea,...

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. View 5164_UV.mw on MapleNet or Download 5164_UV.mw
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## Lets try...

Thank you. I will try it out.

## Hello! Thank you! The result...

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.

## Thanks...

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,...

I found two wrong signs, hopefully now it is correct: PDE := { diff(alpha(u, v), u) = -I*exp(I*u)*beta(u, v), diff(alpha(u, v), v) = -I*exp(-I*v)*beta(u, v), diff(alpha(u, v), u) = I*exp(-I*u)*beta(u, v), diff(alpha(u, v), v) = -I*exp(I*v)*beta(u, v), diff(beta(u, v), u) = -1/4*I*exp(-I*v)*alpha(u, v)-1/2*I*beta(u, v)-1/2*I*exp(-I*(v+u))*beta(u, v), diff(beta(u, v), v) = -1/4*I*exp(-I*u)*alpha(u, v)-1/2*I*beta(u, v)+1/2*I*exp(-I*(v+u))*beta(u, v), diff(beta(u, v), u) = -1/4*I*exp(I*v)*alpha(u, v)-1/2*I*exp(I*(v+u))*beta(u, v)+1/2*I*beta(u, v), diff(beta(u, v), v) = 1/4*I*exp(I*u)*alpha(u, v)+1/2*I*exp(I*(v+u))*beta(u, v)+1/2*I*beta(u, v)}; The funny thing is, that in my Maple output the sign in the 2nd beta-equation diff(beta(u, v), v) = -1/4*I*exp(-I*u)*alpha(u, v)-1/2*I*beta(u, v)+1/2*I*exp(-I*(v+u))*beta(u, v), at -1/2*I*beta(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(u,v)=-I (e)^((I u)) beta(u,v),(∂)/(∂u) alpha(u,v)=(I beta(u,v))/((e)^((I u))),(∂)/(∂u) beta(u,v)=-(1/4 I ((e)^((I u)) alpha(u,v)+2 (e)^((I v)) (e)^((I u)) beta(u,v)+2 beta(u,v)))/((e)^((I v)) (e)^((I u))),(∂)/(∂u) beta(u,v)=-1/4 I ((e)^((I v)) alpha(u,v)+2 (e)^((I v)) (e)^((I u)) beta(u,v)-2 beta(u,v)),(∂)/(∂v) alpha(u,v)=-(I beta(u,v))/((e)^((I v))),(∂)/(∂v) alpha(u,v)=-I (e)^((I v)) beta(u,v),(∂)/(∂v) beta(u,v)=1/4 (-I alpha(u,v) (e)^((I v))-2 I (e)^((I v)) (e)^((I u)) beta(u,v)+2 I beta(u,v))/((e)^((I v)) (e)^((I u))),(∂)/(∂v) beta(u,v)=1/4 I ((e)^((I u)) alpha(u,v)+2 (e)^((I v)) (e)^((I u)) beta(u,v)+2 beta(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,...

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...

Hi! Sorry for being so late now. Here's the code: > > with(PDETools); > PDE := {diff(alpha(u, v), u) = -I*exp(I*u)*beta(u, v), diff(alpha(u, v), v) = -I*exp(-I*v)*beta(u, v), diff(alpha(u, v), u) = I*exp(-I*u)*beta(u, v), diff(beta(u, v), v) = 1/4*I*exp(I*u)*alpha(u, v)+1/2*I*exp(I*(v+u))*beta(u, v)+1/2*I*(beta)(u, v), diff(alpha(u, v), v) = -I*exp(I*v)*beta(u, v), diff(beta(u, v), u) = -1/4*I*exp(-I*v)*alpha(u, v)-1/2*I*beta(u, v)-1/2*I*exp(-I*(v+u))*beta(u, v), diff(beta(u, v), v) = -1/4*I*exp(-I*u)*alpha(u, v)+1/2*I*beta(u, v)+1/2*I*exp(-I*(v+u))*beta(u, v), diff(beta(u, v), u) = -1/4*I*exp(I*v)*alpha(u, v)-1/2*I*exp(I*(v+u))*beta(u, v)-1/2*I*beta(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.
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