The wave equation on the lattice in two and three dimensions

The Klein–Gordon equation on Z2 and the quantum harmonic lattice

“…Unlike in the continuous setting, varying parameters λ 1 , λ 2 is not equivalent to performing a diagonal linear transformation on x because the domain Z 2 and its Fourier dual both lack a dilation symmetry. We show in this paper that the dispersion pattern for (1.1) retains the same topological and geometric structure found in [17] for all values of ω, λ j > 0. In particular there is no choice of parameters that generates exceptional degeneracy or bifurcation of the phase function singularities which determine the dispersive estimate.…”

“…We claim that the methods in [17] apply to the more general case λ 1 , λ 2 > 0, ω = 0 with minimal modification. Specifically, the leading order expression will be (2.24)…”

“…The fundamental solution can still be determined as a superposition of plane waves, with size bounds in the different regimes resulting from stationary phase principles. Isotropic wave equations (λ j = 1, ω = 0) in two and three dimensions were analyzed by Schultz [17], with a curious set of outcomes. In addition to the expected wavefront expanding radially at |y − x| ∼ |t|, there is a secondary region of reduced dispersion traveling at somewhat lower speed.…”

“…The endpoint case ω = 0 corresponds to the discrete wave equation, and it introduces a new type of asymptotic behavior at the boundary of the light cone because the phase function γ(k) develops an absolute-value singularity at the origin. Schultz computed the resulting dispersive estimate in [17] under the further assumption λ 1 = λ 2 . We provide a restatement of these bounds for general λ j in Section 2.2.…”

Diffraction on the Two-Dimensional Square Lattice

“…Together with the paper of Islami and Vainberg [20], this paper explains that, for example, as t → ∞, the solution of the time-dependent problem (1.1) approaches U e iωt , where U solves (1.3). We should also mention the work of Schultz [29], who uses pointwise estimates of the Green's functions for the discrete wave equation to analyze the solution of both linear and nonlinear lattice wave equations in two and three spatial dimensions. Other papers from the numerical analysis literature [4,32] analyze problems similar to (1.3) to determine how closely solutions of the discrete problem approximate solutions of the continuum problem.…”