Hopf-Cole transformation to some systems of partial differential equations (Q5943870)

A system of nonlinear partial differential equations of the form \[ (u_j)_t + \frac{1}{2}\sum_{k=1}^N \sum_{i=1}^j (u_i)_{x_k}(u_{j-i+1})_{x_k} = \frac{\nu}{2} \Delta u_j \] is studied. The authors reduce this system to the Burgers equation in the matrix form and use Hopf-Cole transformation for linearization. As application the initial value problem and initial-boundary value problem with Dirichlet data on the boundary of a cylindrical domain is solved. Also a solution to the initial value problem for more singular \(2 \times 2\) systems in the space of Colombeau distributions is constructed.