Periodic and optical soliton solutions of the quintic complex Swift-Hohenberg equation

Using a direct ansatz approach, we have found a number of periodic zero-velocity analytic solutions of the complex quintic Swift-Hohenberg equation (CSHE). These find application in assorted optical problems. Particular cases of periodic solutions, where the elliptic function modulus equals 1, are various localized solutions of the CSHE. Each of these solutions exists for a certain relation between the parameters of the equation. As a result, they are particular cases of the complete set of periodic and localised solutions which may exist for this equation. In fact, they are multi-parameter families of solutions and they can serve as a seeding set of solutions which could be useful in other optical studies. We have also derived energy and momentum balance equations for the solutions of CSHE and checked that our stationary solutions satisfy the energy balance equation.