Existence and Stability of Traveling Waves for Discrete Nonlinear Schrödinger Equations Over Long Times

Abstract

We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schrödinger equation (DNLS) with cubic nonlinearity, near the continuous limit. We construct a family of solutions close to the continuous traveling waves and prove their stability over long times. Applying a modulation method, we also show that we can describe the dynamics near these discrete traveling waves over long times.

Continuous limits of linear and nonlinear quantum walks

“…We next define the Shannon interpolation (see, e.g. [5]), which is an isometry from H δ to H δ ⊂ L 2 , by…”

Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on <inline-formula><tex-math id="M1">\begin{document}$ h\mathbb{Z} $\end{document}</tex-math></inline-formula>

“…Construction of the modified energies. DNLS is a Hamiltonian differential equation (see [5]) whose Hamiltonian (i.e. its energy) is defined on L 2 pZq by…”

“…An uniform control of these norms with respect to h may be crucial to establish aliasing 1 or consistency estimates. For example, in [5], the existence and the stability of traveling waves is studied near the continuous limit of the focusing DNLS. The discrete Sobolev norms are used to control an aliasing error generated by the variations of the momentum (see Theorem 1.5 of [5]).…”

“…DNLS is a popular model in numerical analysis for the spatial discretization of the cubic nonlinear Schrödinger equation (NLS), given by: (2) iB t u " B 2 x u `ν|u| 2 u, see, for example, [3], [4], [5], [10], [11], [12]. Motivated by the approximation properties of NLS by DNLS, we consider the discrete model near its continuous limit i.e.…”

“…For example, in [5], the existence and the stability of traveling waves is studied near the continuous limit of the focusing DNLS. The discrete Sobolev norms are used to control an aliasing error generated by the variations of the momentum (see Theorem 1.5 of [5]). It is proven that if for all n P N, the discrete Sobolev norm of order n of the solutions of the focusing DNLS can be bounded by t αn , uniformly with respect to h, then DNLS admits solutions whose behavior is similar to traveling waves for times of order h ´β , with β " lim sup n n αn .…”