Localized states and their stability near a combined linear and nonlinear metasurface

We study analytically and numerically the localized states of nonlinear waves propagating along a metasurface (thin defect layer, waveguide) having combined linear and nonlinear properties. In the framework of the nonlinear Schro<spacing diaeresis>dinger equation with delta-functional potential containing both linear and nonlinear spatial perturbations, we find all possible solutions localized near the metasurface in a linear medium, and analyze their conditions of existence. It is shown that the solutions localized near the metasurface are possible for any sign of anharmonicity inside the defect layer in the case of attraction of elementary excitations to the metasurface. However, for the mutual repulsion of elementary excitations inside the defect layer, the localized states can exist only in the case of attractive metasurface. For all possible localized states, the total number of elementary excitations and total energy of the system were calculated. We performed a full analysis of the stability of all localized states both analytically, using the Vakhitov-Kolokolov (VK) criterion and "anti-VK" criterion, and numerically, and found that only localized solutions with attraction of elementary excitations to the metasurface will be stable. The results of direct numerical simulations demonstrated full compliance with the analytical results of the stability study.