Collect all terms containing the same powers of a function in a polynomial

Question

I have a polynomial including powers of [CapitalPhi][[Xi]]

I want to collect all terms containing the same powers of [CapitalPhi][[Xi]]

Then,    I want to equate the coefficients of the terms to $0.$ (It yields to the system of algebraic equations.)

Finally, I want to solve the equation system including the unknowns Subscript[a, 0], Subscript[a, 1], Subscript[b, 1], r. (Other terms such as Subscript[[Lambda], 1], Subscript[[Lambda], 2] are parameters. They can be assigned to $1$)

POLY:=(r - p Subscript[[Lambda], 1] +p^2 Subscript[[Lambda], 2]) (Subscript[a, 0] + Subscript[b, 
                    1]/[CapitalPhi][[Xi]] + Subscript[a, 1] [CapitalPhi][[Xi]]) + 
                 Subscript[[Lambda], 
                  3] (Subscript[a, 0] + Subscript[b, 1]/[CapitalPhi][[Xi]] + 
                    Subscript[a, 1] [CapitalPhi][[Xi]])^3 - 
                 Subscript[[Lambda], 
                  2] (2 Subscript[a, 
                     1] [CapitalPhi][[Xi]] (2 + [CapitalPhi][[Xi]]^2) + 
                    Subscript[b, 
                     1] (-((2 (2 + [CapitalPhi][[Xi]]^2))/[CapitalPhi][[Xi]]) + (
                       2 (2 + [CapitalPhi][[Xi]]^2)^2)/[CapitalPhi][[Xi]]^3));
        
CoefficientList[Expand[POLY], [CapitalPhi][[Xi]]^_]

mikado · Answer

This generates equations you are trying to solve.
Series[POLY /. Φ[ξ] -> s, {s, 0, 8}] == 0 // LogicalExpand // FullSimplify

I leave the solution of these equations to you.
Notes:

Change of variable to s is likely to make the algebra more reliable
Series expansion generates the coefficients for s that I think you are asking for (no harm in asking for an unnecessarily high order of expansion)
LogicalExpand turns these into a set of equations, one per term.
Seriously, I'd avoid using subscripts in this sort of manipulation (e.g. use Subscript[λ, 3]->λ3 to replace them (and reverse this after you have the answer you want to present)

Roman · Answer

Is SolveAlways what you need?
POLY = (r - p λ1 + p^2 λ2) (a0 + b1/Φ[ξ] + a1 Φ[ξ]) +
       λ3 (a0 + b1/Φ[ξ] + a1 Φ[ξ])^3 -
       λ2 (2 a1 Φ[ξ] (2 + Φ[ξ]^2) +
       b1 (-((2 (2 + Φ[ξ]^2))/Φ[ξ]) + (2 (2 + Φ[ξ]^2)^2)/Φ[ξ]^3));

SolveAlways[POLY == 0, Φ[ξ]]
(*    {{a0 -> 0, a1 -> 0, b1 -> 0},
       {r -> p λ1, λ2 -> 0, λ3 -> 0},
       {r -> p λ1 - p^2 λ2 - a0^2 λ3, a1 -> 0, b1 -> 0},
       {r -> p λ1 - p^2 λ2, a1 -> 0, b1 -> 0, λ3 -> 0},
       {r -> p λ1 - a0^2 λ3, a1 -> 0, λ2 -> 0, b1 -> 0},
       {r -> p λ1 - a0^2 λ3, b1 -> 0, λ2 -> 0, a1 -> 0},
       {r -> p λ1 - a0^2 λ3, λ2 -> 0, a1 -> 0, b1 -> 0},
       {r -> 1/2 (2 p λ1 - 2 a0^2 λ3 - a0^2 p^2 λ3),
           a1 -> 0, λ2 -> (a0^2 λ3)/2, b1 -> 0},
       {r -> 1/2 (2 p λ1 + 4 a1^2 λ3 - a1^2 p^2 λ3),
           b1 -> 0, λ2 -> (a1^2 λ3)/2, a0 -> 0},
       {r -> 1/8 (8 p λ1 + 4 b1^2 λ3 - b1^2 p^2 λ3),
           a1 -> 0, λ2 -> (b1^2 λ3)/8, a0 -> 0},
       {a0 -> 0, a1 -> -(b1/2),
           r -> 1/8 (8 p λ1 + 16 b1^2 λ3 - b1^2 p^2 λ3),
           λ2 -> (b1^2 λ3)/8},
       {a0 -> 0, a1 -> b1/2,
           r -> 1/8 (8 p λ1 - 8 b1^2 λ3 - b1^2 p^2 λ3),
           λ2 -> (b1^2 λ3)/8}}                              *)

Collect all terms containing the same powers of a function in a polynomial

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