Discrete Solitary Waves in Systems with Nonlocal Interactions and the Peierls-Nabarro Barrier

Abstract: We study a class of discrete focusing nonlinear Schr{\"o}dinger equations (DNLS) with general nonlocal interactions. We prove the existence of onsite and offsite discrete solitary waves, which bifurcate from the trivial solution at the endpoint frequency of the continuous spectrum of linear dispersive waves. We also prove exponential smallness, in the frequency-distance to the bifurcation point, of the Peierls-Nabarro energy barrier (PNB), as measured by the difference in Hamiltonian or mass functionals evaluated on the onsite and offsite states. These results extend those of the authors for the case of nearest neighbor interactions to a large class of nonlocal short-range and long-range interactions. The appearance of distinct onsite and offsite states is a consequence of the breaking of continuous spatial translation invariance. The PNB plays a role in the dynamics of energy transport in such nonlinear Hamiltonian lattice systems. Our class of nonlocal interactions is defined in terms of coupling coefficients, $J_m$, where $m\in\mathbb{Z}$ is the lattice site index, with $J_m\simeq m^{-1-2s}, s\in[1,\infty)$ and $J_m\sim e^{-\gamma|m|},\ s=\infty,\ \gamma>0,$ (Kac-Baker). For $s\ge1$, the bifurcation is seeded by solutions of the (effective / homogenized) cubic focusing nonlinear Schr{\"o}dinger equation (NLS). However, for $1/4<s<1$, the bifurcation is controlled by the fractional nonlinear Schr{\"o}dinger equation, FNLS, with $(-\Delta)^s$ replacing $-\Delta$. The proof is based on a Lyapunov-Schmidt reduction strategy applied to a momentum space formulation. The PN barrier bounds require appropriate uniform decay estimates for the discrete Fourier transform of DNLS discrete solitary waves. A key role is also played by non-degeneracy of the ground state of FNLS, recently proved by Frank, Lenzmann & Silvestre.