Nonlinear matrix differential equation

I am working with the following differential equation:

$$\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$$

which is essentially in matrix notation:

$$\dot{\mathbf{x}} = A\mathbf{x} + \mathrm{diag}(\mathbf{x)}B\mathbf{x}$$

with $x\in \mathbb{R}^n$ and $A,B\in \mathbb{R}^{n\times n}$ and $B^T=-B$ and A normal.

I wondered if you had any idea how to approach the nonlinear part? I found the paper (1) which gives some hints for approximations, but essentially it is of no help. Maybe you know how to deal with it?

(1) Elliot W.Montroll: On coupled Rate Equations with Quadratic Nonlinearities

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• Might Mathematics be better suited for this math question? – Kyle Kanos May 13 '15 at 14:06
• Kyle Kanos: I also posed the question at mathematical places, but I am quite sure that physicists might tend to bring more pragmatic ideas than proofs of existence. – varantir May 13 '15 at 14:09
• The problem is that this is purely a mathematical question whereas we prefer our questions to be actually about the physics. It's simply not a good fit for our site, IMO. – Kyle Kanos May 13 '15 at 14:32
• Dear varantir. In general, it is frown upon to cross-post simultaneously, because it may waste potential answerer's time. As a minimum OP should mention the cross-post (on both sites!). The preferred procedure is to not cross-post, and if the post hasn't received an acceptable answer after, say, a couple of days, then OP could flag for migration. – Qmechanic May 13 '15 at 14:41
• This seems tough with no prior knowledge of $A,B$. I would think that a lot more could be said if, for example, you knew that $A$ and $B$ were invertible and symmetric. – Ian May 13 '15 at 22:42