Rogue-wave solutions for a discrete Ablowitz–Ladik equation with variable coefficients for an electrical lattice

Abstract

We investigate a discrete Ablowitz–Ladik equation with variable coefficients, which models the modulated waves in an electrical lattice. Employing the similarity transformation and Kadomtsev–Petviashvili hierarchy reduction, we obtain the rogue-wave solutions in the Gram determinant form under certain variable-coefficient constraints. We graphically study the rogue waves with the influence of the coefficient of tunnel coupling between the sites, $$|\varLambda (t)|$$ , time-modulated effective gain/loss term, $$\gamma (t)$$ , space–time-modulated inhomogeneous frequency shift, $$v_n(t)$$ ( $$n=1,2,\ldots $$ ), and lattice spacing, h, where t is the scaled time. Increasing value of h leads to the decrease in the rogue waves’ amplitudes. Properties of the rogue waves with $$|\varLambda (t)|$$ as the polynomial, sinusoidal, hyperbolic and exponential functions are discussed, respectively. The monotonically increasing, monotonically decreasing, periodic and Gaussian backgrounds are, respectively, displayed with the different $$\gamma (t)$$ . The first-order rogue wave exhibits one hump and two valleys, and the second-order rogue waves exhibit three humps and one highest peak. The third-order rogue waves with the six humps and one highest peak are also presented.