Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations

Abstract

For a number of nonlocal nonlinear equations such as nonlocal, nonlinear Schrödinger equation (NLSE), nonlocal Ablowitz-Ladik (AL), nonlocal, saturable discrete NLSE (DNLSE), coupled nonlocal NLSE, coupled nonlocal AL, and coupled nonlocal, saturable DNLSE, we obtain periodic solutions in terms of Jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions. Remarkably, in all the six cases, we find that unlike the corresponding local cases, all the nonlocal models simultaneously admit both the bright and the dark soliton solutions. Further, in all the six cases, not only the elliptic functions dn(x, m) and cn(x, m) with modulus m but also their linear superposition is shown to be an exact solution. Finally, we show that the coupled nonlocal NLSE not only admits solutions in terms of Lamé polynomials of order 1 but also admits solutions in terms of Lamé polynomials of order 2, even though they are not the solution of the uncoupled nonlocal problem. We also remark on the possible integrability in certain cases.