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We introduce the novel concept of a bound state in the continuum (BIC) for a binary lattice satisfying the [ABSTRACT FROM AUTHOR]

• A fractional version of 2D discrete Laplacian is inserted into the DNLS equation. • Spectrum of linear waves and root mean square displacement are computed in closed form. • Insertion of single general nonlinear surface impurity at the edge and at a corner. • Use of Green functions: method of images. • Compute the mode energy and bound state profile. We study the formation of localized modes around a generalized nonlinear impurity which is located at the boundary of a semi-infinite square lattice, and where we replace the standard discrete Laplacian by a fractional one, characterized by a fractional exponent 0 < α < 1 where α = 1 marks the standard, non-fractional case. We specialize to two impurity cases: impurity at an "edge" and impurity at a "corner" and use the formalism of lattice Green functions to obtain in closed form the bound state energy and its mode amplitude. It is found that, for any fractional exponent and for impurity strengths above a certain threshold, there is always a single bound state for the linear impurity, while for the nonlinear (cubic) case, up to two bound states are possible. At small fractional exponents, the energy of the impurity mode becomes directly proportional to the impurity strength. [ABSTRACT FROM AUTHOR]

• A fractional version of discrete Laplacian is inserted into the DNLS equation. • Spectrum of linear waves and root mean square displacement computed in closed form. • Insertion of single general nonlinear impurity. • Use of Green functions to compute the mode energy and profile and the transmission of plane waves. • Effect of fractional exponent on all of the above phenomenology. We use a lattice Green function approach to study the stationary modes of a linear/nonlinear (Kerr) impurity embedded in a periodic one-dimensional lattice where we replace the standard discrete Laplacian by a fractional one. The energies and the mode profiles are computed in closed form, for different fractional exponents and different impurity strengths. The energies of the impurity mode lie outside the linear band whose bandwidth decreases steadily as the fractional exponent decreases. For any fractional exponent values, there is always a single bound state for the linear impurity while for the nonlinear (Kerr) case, up to two bound states are possible, for impurity strengths above certain threshold. The energy of the linear mode (or that of the upper energy nonlinear one), becomes directly proportional to the impurity strength at large impurity strengths. The transmission of plane waves is also computed in closed form for several fractional exponents, and various impurity strengths. We observe that fractionality tends to increase the overall transmission. The selftrapping transition for the nonlinear impurity shifts to lower nonlinearity values as the fractional exponent is decreased. In both cases, linear and nonlinear, we observe a form of trapping at zero impurity strength, which can be explained by the near-degeneracy of the spectrum in the limit of a small fractional exponent. [ABSTRACT FROM AUTHOR]

We introduce a novel concept of surface bound states in the continuum, i.e., surface modes embedded into the linear spectral band of a discrete lattice. We suggest an efficient method for creating such surface modes and the local bounded potential necessary to support the embedded modes. We demonstrate that the surface embedded modes are structurally stable, and the position of their eigenvalues inside the spectral band can be tuned continuously by adding weak nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract: We study localized modes on a single magnetic impurity positioned in the bulk or at the surface of a one-dimensional chain, in the presence of a magnetic field B acting at the impurity site. The strong on-site nonlinear interaction U between two electrons of opposite spin at the impurity site, modelled here as a nonlinear local term, and the presence of the external field induce a strong correlation between parallel and antiparallel spin bound states. We find that, for an impurity in the bulk, a localized vector mode (with up and down spin components) is always possible for any given value of U and B, while for a surface impurity, a minimum value of both, U and B is needed to create a vector mode. In this case, up to two localized modes are possible, but only one of them is stable. The presence of the surface seems to destabilize the bulk mode in the parameter region , creating a “forbidden strip” region in parameter space, bounded by and , approximately. [Copyright &y& Elsevier]