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Question: why i can't remove the the denominatore of system?

Question:why i can't remove the the denominatore of system?

Math-dashti 135 Maple 2024
denominator duplicate-account
5 hours ago
1

is work for some equation but sometime is make problem and again make it problem for me where is problem why the denominator of second equation still remain while i want to remove it i times by denominator but still not worked

in my orginal ode i did change the place diff(V(xi),xi)=Omega(xi) maybe make problem ...or not

like this equation but the equation is different

> 

restart
> 

with(PDEtools)
> 

with(plots)
> 

with(plots):
> 

with(DEtools):
> 

undeclare(prime, quiet)
> 

with(LinearAlgebra)
> 

declare(u(x, t), quiet); declare(U(xi), quiet); declare(V(xi), quiet); declare(Omega(xi), quiet)
> 

ode := -(8*(1/4+k^2*alpha[1]^2+((1/2)*w^2*alpha[2]-k)*alpha[1]))*V(xi)*(diff(Omega(xi), xi))+(4*k^2*alpha[1]^2+1+(2*w^2*alpha[2]-4*k)*alpha[1])*Omega(xi)^2+8*w^2*(-alpha[4]*V(xi)^2+k^2*alpha[1]+(1/2)*w^2*alpha[2]-alpha[3]*V(xi)-k)*alpha[2]*V(xi)^2 = 0

-8*(1/4+k^2*alpha[1]^2+((1/2)*w^2*alpha[2]-k)*alpha[1])*V(xi)*(diff(Omega(xi), xi))+(4*k^2*alpha[1]^2+1+(2*w^2*alpha[2]-4*k)*alpha[1])*Omega(xi)^2+8*w^2*(-alpha[4]*V(xi)^2+k^2*alpha[1]+(1/2)*w^2*alpha[2]-alpha[3]*V(xi)-k)*alpha[2]*V(xi)^2 = 0

 (1)
> 

NULL
> 

raw := DEtools[convertsys]({ode}, {}, Omega(xi), xi, s, QP, QP)[1..2];

[[QP[1] = -(1/8)*(-(4*k^2*alpha[1]^2+1+(2*w^2*alpha[2]-4*k)*alpha[1])*s[1]^2-8*w^2*(-alpha[4]*V(xi)^2+k^2*alpha[1]+(1/2)*w^2*alpha[2]-alpha[3]*V(xi)-k)*alpha[2]*V(xi)^2)/((1/4+k^2*alpha[1]^2+((1/2)*w^2*alpha[2]-k)*alpha[1])*V(xi))], [s[1] = Omega(xi)]]

 (2)

Extract the denominator and scale the right hand sides by it

> 

den:=denom(eval(QP[2],raw[1]));
raw_eta:=map(q->rhs(q)*den,raw[1]);

1

  

[-(1/8)*(-(4*k^2*alpha[1]^2+1+(2*w^2*alpha[2]-4*k)*alpha[1])*s[1]^2-8*w^2*(-alpha[4]*V(xi)^2+k^2*alpha[1]+(1/2)*w^2*alpha[2]-alpha[3]*V(xi)-k)*alpha[2]*V(xi)^2)/((1/4+k^2*alpha[1]^2+((1/2)*w^2*alpha[2]-k)*alpha[1])*V(xi))]

 (3)

Back to the real transformed variables, which are now in terms of eta.

> 

rhs_eta := eval(raw_eta, {s[1] = phi(eta), s[2] = y(eta)})

[2*y(eta)*(4*k^2*alpha[1]^2+2*w^2*alpha[1]*alpha[2]-4*k*alpha[1]+1)*phi(eta), -(1/4)*(-(4*k^2*alpha[1]^2+1+(2*w^2*alpha[2]-4*k)*alpha[1])*y(eta)^2-8*w^2*(-alpha[4]*phi(eta)^2+k^2*alpha[1]+(1/2)*w^2*alpha[2]-alpha[3]*phi(eta)-k)*alpha[2]*phi(eta)^2)*(4*k^2*alpha[1]^2+2*w^2*alpha[1]*alpha[2]-4*k*alpha[1]+1)/(1/4+k^2*alpha[1]^2+((1/2)*w^2*alpha[2]-k)*alpha[1])]

 (4)

Find equilibrium points - one is at the origin; the others are a complicated mess.

> 

equilibria := [solve(rhs_eta, {phi(eta), y(eta)}, explicit)]; nops(%)

3

 (5)

Eq 9.

> 

de1 := diff(phi(eta), eta) = rhs_eta[1]; de2 := diff(y(eta), eta) = rhs_eta[2]

diff(phi(eta), eta) = 2*y(eta)*(4*k^2*alpha[1]^2+2*w^2*alpha[1]*alpha[2]-4*k*alpha[1]+1)*phi(eta)

  

diff(y(eta), eta) = -(1/4)*(-(4*k^2*alpha[1]^2+1+(2*w^2*alpha[2]-4*k)*alpha[1])*y(eta)^2-8*w^2*(-alpha[4]*phi(eta)^2+k^2*alpha[1]+(1/2)*w^2*alpha[2]-alpha[3]*phi(eta)-k)*alpha[2]*phi(eta)^2)*(4*k^2*alpha[1]^2+2*w^2*alpha[1]*alpha[2]-4*k*alpha[1]+1)/(1/4+k^2*alpha[1]^2+((1/2)*w^2*alpha[2]-k)*alpha[1])

 (6)
> 

PDEtools:-ConservedCurrents({de1, de2}, [phi(eta), y(eta)]); P1 := -(1/2)*op(1, rhs(op(%)))

[_J[eta](eta, phi(eta), y(eta)) = f__1((1/3)*(2*w^2*alpha[2]*phi(eta)^4*alpha[4]+3*w^2*alpha[2]*phi(eta)^3*alpha[3]-6*(k^2*alpha[1]+(1/2)*w^2*alpha[2]-k)*alpha[2]*w^2*phi(eta)^2+3*((1/2)*w^2*alpha[1]*alpha[2]+(k*alpha[1]-1/2)^2)*y(eta)^2)/(((1/2)*w^2*alpha[1]*alpha[2]+(k*alpha[1]-1/2)^2)*phi(eta)), (1/2)*3^(1/2)*Intat(1/((-96*(-(1/3)*w^2*alpha[2]*phi(eta)^4*alpha[4]-(1/2)*w^2*alpha[2]*phi(eta)^3*alpha[3]+(k^2*alpha[1]+(1/2)*w^2*alpha[2]-k)*alpha[2]*w^2*phi(eta)^2-_a*(k^2*alpha[1]+(1/2)*w^2*alpha[2]-(1/3)*_a^2*alpha[4]-(1/2)*_a*alpha[3]-k)*w^2*alpha[2]*phi(eta)-(1/2)*((1/2)*w^2*alpha[1]*alpha[2]+(k*alpha[1]-1/2)^2)*y(eta)^2)*_a*((1/2)*w^2*alpha[1]*alpha[2]+(k*alpha[1]-1/2)^2)/phi(eta))^(1/2)*_a), _a = phi(eta))+eta)]

  

-(1/6)*(2*w^2*alpha[2]*phi(eta)^4*alpha[4]+3*w^2*alpha[2]*phi(eta)^3*alpha[3]-6*(k^2*alpha[1]+(1/2)*w^2*alpha[2]-k)*alpha[2]*w^2*phi(eta)^2+3*((1/2)*w^2*alpha[1]*alpha[2]+(k*alpha[1]-1/2)^2)*y(eta)^2)/(((1/2)*w^2*alpha[1]*alpha[2]+(k*alpha[1]-1/2)^2)*phi(eta))

 (7)

NULL

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