This paper is concerned with the characterization
of the weak limits (delta waves) associated to the Cauchy
problem for the Burgers' equation and the inviscid Burgers' equation
with strongly singular
initial data in the form of a regularization by smooth mollifiers of sums
of derivatives of Dirac measures.
By means of Laplace's method we give precise asymptotic
expansion of the solutions. Then we apply
these asymptotics in order to
classify completely all possible delta waves under a suitable nondegeneracy
condition on some mollifiers regularizing the leading singular term of the
initial data.
We propose also certain stability results for the weak limits under
suitable perturbations of the initial data.