Stability of Intrinsic Localized Modes on the Lattice with Competing Power Nonlinearities

Abstract. We study the discrete nonlinear Schrödinger equation with competing powers [Formula: see text] satisfying [Formula: see text]. The physically relevant cases are given by [Formula: see text], [Formula: see text], and [Formula: see text]. In the anticontinuum limit, all intrinsic localized modes are compact and can be classified by their codes, which record one of two nonzero (smaller and larger) states and their sign alternations. By using the spectral stability analysis, we prove that the codes for larger states of the same sign are spectrally and nonlinearly (orbitally) stable, whereas the codes for smaller states of the alternating signs are spectrally stable but have eigenvalues of negative Krein signature. We also identify numerically the spectrally stable codes which consist of stacked combinations of the sign-definite larger states and the sign-alternating smaller states.