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I am currently dealing with a nonlinear system of coupled equations. In fact I had performed a perturbation approach for this system which is highly nonlinear. Thanks to first step of the perturbative approach I could reach eigenvalues in the "linear case". Right now I want to move to next order of the perturbative approach, thus I must deal with nonlinear terms. Even if the system is nonlinear, physically speaking eigenvalues must exist, but I have no ideas how I could compute them.

Of course the system has many equations, but in order to simplify it a bit, I only consider two coupled equations:

Ω_1*E_1 + K_{11} |E_1|^2 E_1 + K_{12} |E_2|^2 E_1 = 0

Ω_2*E_2 + K_{21} |E_1|^2 E_2 + K_{22} |E_2|^2 E_2 = 0

Ω_1, Ω_2 are eigenvalues in the linear case. In nonlinear case, the coupling is performed by K_{ij} terms.

I wonder if someone knows a analytical approach of this problem? And if there is no such method, is there an iterative or numerical approach which leads to computable solutions?

Best

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# Eigenvalues of nonlinearly coupled equations

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