Partial-limit solutions and rational solutions with parameter for the Fokas-Lenells equation

A partial-limit method is developed to understand solutions related to real eigenvalues for the Fokas-Lenells equation. By applying a partial-limit procedure to soliton solutions of the Fokas-Lenells equation, new multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop vertical bar u vertical bar(2), the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly changing amplitudes. The partial-limit procedure may be extended to other integrable systems and generate new solutions.