Asymptotics of the Head Wave in the Cauchy Problem for a Difference Scheme Corresponding to the Two-Dimensional Wave Equation with Localized Initial Data

An asymptotics for the head wave in the Cauchy problem for the pseudodifferential equation corresponding to a difference scheme for the wave equation with localized initial data is constructed. Dispersive effects appear in the head wave, depending on the relation between the parameters of the problem. In the case of strong dispersive effects, the uniform asymptotic behavior of the head wave in a sufficiently large neighborhood of the leading front is constructed. Moreover, such an asymptotics can be represented in terms of Jacobi theta-functions in the case where the initial function has the form of a Gaussian exponential.