I am facing the following non-linear matrix differential equation. $V$ is a symmetric $n\times n$ matrix, $u=(1,1, \dots,1) \in \mathbb{R}^n$ a row vector and $a$ is a constant. $$\frac{\mathrm{d}V}{\mathrm{d}t}(t)=aV^2(t)\frac{u^{T}u}{uV^2(t)u^T}V^2(t) $$ In bra-ket notation : $$ \frac{\mathrm{d}V}{\mathrm{d}t}(t)=a\frac{V^2(t)|u><u|V^2(t)}{<u|V^2(t)|u>}$$
I tried to use the diagonalization of $V$ : $V(t)=\sum_i\lambda_i(t)v_i^T(t)v_i(t)$ to get equations over $\lambda_i$ and $v_i$ but it doesn't seems to help...
I also tried to multiply the equation by $u$ on the right to get a differential equation on $w(t)=V(t)u$ : $$\frac{\mathrm{d}w}{\mathrm{d}t}(t)=aV(t)w(t) $$ that is solvable by the ordered exponential: $$w(t)=\mathcal{T}\lbrace \exp(\int_0^taV(t')\mathrm{d}t)\rbrace w(0) $$ But I don't think that introducing ordered exponential will simplify the problem.
As someone already faced such equation ? Is it solvable and do you have a clue to solve it ?