Eigenvalues of nonlinearly coupled equations

[nbachela]

Hi everyone,

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

 

[mfb]

Are those factors real or complex numbers?
If yes, it looks like you can divide the first equation by E_1 and the second one by E_2. Afterwards, you have linear equations in the squared values.
If no, what are they?

 

[nbachela]

Hi,

Thanks to your last reply I just realize I made a mistake. The Ω are not constant values. In fact the system correspond to coupled differential equations:

dE_1/dt+ω_1*E_1 + K_{11} |E_1|^2 E_1 + K_{12} |E_2|^2 E_1 = 0
dE_2/dt+ω_2*E_2 + K_{21} |E_1|^2 E_2 + K_{22} |E_2|^2 E_2 = 0

Sorry for the mistake.


Best

 

[mfb]

Okay, and what are you looking for now? General solutions E1(t), E2(t)? Some special solutions? E1, E2 where the derivatives vanish? The last case could be reduced to the suggestion in my previous post.

 

[nbachela]

In fact I would like to compute the eigensolutions of this system. I am not sure I can even talk about eigenvalue for a non linear system.

To tell the whole story, I need to compute mode of a particular multimode laser, so this eigensolutions have a physical meaning (they corresponds to emitted modes). In fact my system is much more complicated, but I have simplified it a bit. In real, it is not possible to divide by E_i in all the equations because of coupling terms. If I write the system for 3 coupled modes, it would look like this:

dE_1/dt+ω_1*E_1 + K_{11} |E_1|^2 E_1 + K_{12} |E_2|^2 E_1 + K_{13} |E_3|^2 E_1 + K_{123} E_1*E_2*E_3= 0
dE_2/dt+ω_2*E_2 + K_{21} |E_1|^2 E_2 + K_{22} |E_2|^2 E_2 + K_{23} |E_3|^2 E_2 + K_{231} E_1*E_2*E_3 = 0
dE_3/dt+ω_2*E_2 + K_{31} |E_1|^2 E_3 + K_{32} |E_2|^2 E_3 + K_{33} |E_2|^3 E_3 + K_{312} E_1*E_2*E_3= 0



I would like to know if there is an analytical or iterative way to solve such system?

 

[mfb]

Iterative way: Sure. For each step in time, calculate dE_i/dt, let it evolve to the next step and repeat.
Those expanded equations still have E_i as common factor in equation i.