Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices

Abstract

Abstract. We construct small amplitude breathers in 1D and 2D Klein-Gordon infinite lattices. We also show that the breathers are well approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the Klein Gordon lattice and the discrete Non Linear Schrödinger lattice. The proof is based on a Lyapunov-Schmidt decomposition and continuum approximation techniques introduced in [9], actually using its main result as an important lemma.

Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

“…be an initial data. Then there exists a unique global solution A(τ ) of the discrete nonlinear Schrödinger equation (4)…”

“…Proof 3. To prove this lemma, we can use the result from Lemma 1 as well as the property of Banach algebra in ℓ 2 (Z N ), such that from the global existence and smoothness of the solution A(τ ) of the discrete nonlinear Schrödinger equation (4) in Lemma 1, we obtain (19). 6…”

“…where A j τ (0) can be obtained from the Schrödinger equation (4). In this way, the error y(t) = u j (t) − X j (t) will satisfy the initial condition y(0) ℓ 2 = 0.…”

“…The method is usually referred to as the rotating wave approximation. If one is interested in solution profiles with a much larger length scale than the typical distance between the lattices, they can aim for a continuous approximation and obtain nonlinear Schrödinger equations (see, e.g., [14,13,4]). When one is instead interested in waves with the same scale of the typical lattice distance, one will obtain an approximation in the form of a discrete nonlinear Schrödinger equation (see, e.g., [19,11,7,15,16,12,23]) with the corresponding wave properties quite distinct from those in the continuous limit.…”

On the nonexistence of degenerate phase-shift multibreathers in Klein–Gordon models with interactions beyond nearest neighbors

“…and provides explicit (though not sharp) estimates of the approximation of asymmetric vortices in (1) with the corresponding phase-shift discrete solitons in (7) (cf. with Theorem 2.1 of [3]). The price one has to pay for the use of this strategy lies in the restrictions in the regime of parameters for which the models (1) are well approximated by the corresponding averaged normal forms (7).…”