Degenerate perturbation theory to nonlinear equation

Question

I want to use perturbation theory to find the steady-state solution to the following nonlinear equation: $$ x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0, $$ where $i=1\cdots N$ and $\epsilon$ is a small parameters. To apply perturbation theory, we expand $x_i = x_i^{(0)} + \epsilon x_i^{(1)} + \mathcal{O}(\epsilon^2) $.

• The zeroth order solution is given by $$ \sum_{i}\left[x_{i}^{\left(0\right)}\right]^{2} = a, $$ which gives a degenerate of solutions.
• Expanding it to first order in $\epsilon$ gives $$ 2 (x_i^{(0)})^2 x_i^{(1)} + 2x_i^{(0)}\sum_{j\neq i}x_j^{(0)}x_j^{(1)} + \sum_{j\neq i}J_{ij}x^{(0)}_j=0 , $$ but then the solution to $x_i^{(1)}$ involves other $x_j^{(1)}$.
  • I think this issue arises because we have a degenerate of solutions to the unperturbed equation.
  • Is there a way to apply degenerate perturbation theory to obtain an explicit form for $x_i^{(1)}$ ?.

I'm interested in seeing how the perturbations lift the degeneracy of solutions.

Answer 1

Rewrite the system of equations in the following clearer form: $$ \vec{x}\left[\left(|x|^{2}-a\right)I+\epsilon J\right]=0. $$ Then there are 2 possible cases

1. $\left[\left(|x|^{2}-a\right)I+\epsilon J\right]=0.$ This only occurs at $\epsilon =0$ and the solution $|x|^2 =a$ coincides with your original solution, and

2. $\left[\left(|x|^{2}-a\right)I+\epsilon J\right]$ has a zero eigenvectior, and then $\vec x$ is proportional to it. This can be acheived by setting setting $|x|^2-a =-\epsilon\lambda$ where $\lambda$ is an eigenvalue of $J$. At $\epsilon=0$ the solution reverts to the unperturbed problem, but $\vec x$ is again degenerate.

So as long as one can find such an eigenvalue, a second "$\epsilon$" dependent solution exists.