Existence and uniqueness of solutions of nonlinear evolution systems of n-th order partial differential equations in the complex plane

We prove existence and uniqueness results for quasilinear systems of partial differential equations of the form ∂th − ∂ y h = g2 ( y, t, {∂ yh} j=0 ) ∂ y h + g1 ( y, t, {∂ yh} j=0 ) + r(y, t); h(y, 0) = hI(y) for sufficiently large y in a sector in the complex plane, under certain analyticity and decay conditions on h, hI ,g1,g2 and r such that the nonlinearity is formally small as y → ∞. Due to the type of nonlinearity in the highest derivatives and the divergence of formal series solutions owed to the singularity of the system at infinity we needed to use new techniques (based on Borel-Laplace duality) to control the perturbative terms. Our methods of proof and norms used are of a constructive nature. The results given can be used to justify the formal asymptotic expansion of solutions in an appropriate sector and conversely to reconstruct actual solutions from formal series ones.