We consider a discrete nonlinear Klein–Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein–Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

International Press of Boston - publishers of scholarly mathematical and scientific journals and books

In this paper, we study several aspects of the modified Geroch functional of n-dimensional Einstein submanifold Σ ⁿ embedded in an (n + 1)-dimensional Einstein manifold M ⁿ⁺¹,Σ ⁿ ⊂ M ⁿ⁺¹. We parameterized the embedding by a parameter t and using the first variation of MGF, we show that the submanifold Σ ⁿ with non-negative mean curvature H ≥ 0 is a critical point of MGF if and only if is minimal or umbilic.