I am trying to find the solution for the following set of differential equations:
Find $q(u) = [q_1(u), \ldots, q_n(u)]^\top$ where $u \in \mathbb{R}^n$ and $n \in \mathbb{N}$ such that $$ \sum_{k=1}^n \frac{\partial q_i}{\partial u_k}\,\frac{\partial q_j}{\partial u_k} = q_i\, \delta_{ij} - q_i\,q_j \qquad\text{ s.t. }\quad \begin{cases}q_i(u) \geq 0 \,\,\text{ for }\,\, i \in [n]\\ \sum_{i=1}^n q_i(u) \leq 1\\ q(0) = \frac{1}{n+1}\,\mathbf{1}\end{cases}\, , $$ where $\delta_{ij}$ is the Kronecker delta function. The solution is possibly related to partition of unity where $q_{n+1} := 1 - \sum_{i=1}^n q_i(u)$.
For $n=1$, the solution is simply $q_1(u_1) = \frac{1 + sin(u_1)}{2}$ and $\, [q_1(u_1), 1-q_1(u_1)]^\top$ forms a partition of unity. However, I am having trouble generalizing the 1-dimensional case to $n > 1$.