Stability of solitary waves in finite Ablowitz–Ladik lattices

Abstract

We discuss the effects of different types of boundary conditions (b.c.) on the linear stability of a solitary wave in a finite-length dynamical lattice described by the Ablowitz–Ladik (AL) model. Types of b.c. considered are 'fixed' (Dirichlet), no-flux (Neumann) and free as well as periodic b.c. The behaviour of eigenvalues around a stationary nonlinear wave consistent with several types of b.c. is studied analytically and numerically. The translational eigenvalues are found to move away from the origin. It is shown that for b.c. of the 'fixed' type, these eigenvalues bifurcate into a neutrally stable (oscillatory) kind, while the no-flux and free b.c. lead to an exponentially weak instability. The 'rotational' eigenmodes (those related to the gauge invariance of the lattice) strictly remain at the origin, as gauge invariance is not broken by the linear homogeneous b.c. These (numerical) results are compared to the analytical and semi-analytical predictions obtained within the approximation of the images generated by the solitary wave beyond the lattice edges. A potential effect of continuous-spectrum band-edge bifurcations on stability is also evaluated, with the conclusion that that these bifurcations cannot destabilize the 'soliton'. For periodic b.c., the translational invariance is preserved, and so is the Lax-pair structure, sustaining the integrable nature of the problem. The AL model was chosen so that the effects of boundary conditions are monitored in the absence of non-integrable discreteness. Finally, the possibility of the generation of multipulse configurations, due to the interplay of finiteness effects with the exponential tail–tail interaction of the pulses, is examined.