Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation

It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data λ −1 a 0 (λx . η, x)r 1 (η), λ > 0, x, η ∈ R n , η ¬= 0. Here r 1 (η) is a characteristic vector, and a 0 (σ, x) is a smooth scalar function of compact support. Under the additional requirements that n = 2 or 3 and that a 0 (σ, x) have the vanishing mean with respect to σ, it is shown that a genuine solution exists in a time interval independent of λ, and that the formal solution is asymptotic to the genuine solution as λ → ∞