Intrinsic localized modes for DNLS equation with competing nonlinearities: Bifurcations

We study nonlinear excitations described by Discrete Nonlinear Schrödinger (DNLS)-type equations with so-called competing nonlinearities, i.e., the nonlinearities that consist of two power terms with coefficients of a different sign. A key feature of these models is the presence of the two governing parameters: α, which characterizes the coupling between lattice sites, and γ, which quantifies the balance between competing terms. Our study focuses on intrinsic localized modes (ILMs)-the solutions that exhibit spatial localization over a few lattice sites. We consider three versions of the DNLS equation: cubic-quintic, quadratic-cubic, and cubic-quartic models. The last two models are widely discussed nowadays in Bose-Einstein condensate theory. Our approach employs numerical continuation from the anti-continuum limit where the coupling between the lattice sites is neglected (α=0). We analyze α-dependent branches of the basic ILMs and their bifurcations when γ varies, focusing on the features that are common for all the three models. We present comprehensive tables of bifurcations for the ILMs and found that they are common for all the three models, up to bifurcation values. We paid special attention to the branches of ILMs that connect highly discrete and continuous limits of a DNLS equation (called here ∞-branches). We provide numerical evidence that there are exactly two (up to symmetries) ∞-branches for γ∈(0;γ∗), where γ∗ is a threshold value. When γ varies within this range, these branches undergo a sequence of bifurcations that is universal for all the three cases.