Painlevé Analysis and a Solution to the Traveling Wave Reduction of the Radhakrishnan — Kundu — Lakshmanan Equation

This paper considers the Radhakrishnan — Kundu — Laksmanan (RKL) equation to analyze dispersive nonlinear waves in polarization-preserving fibers. The Cauchy problem for this equation cannot be solved by the inverse scattering transform (IST) and we look for exact solutions of this equation using the traveling wave reduction. The Painlevé analysis for the traveling wave reduction of the RKL equation is discussed. A first integral of traveling wave reduction for the RKL equation is recovered. Using this first integral, we secure a general solution along with additional conditions on the parameters of the mathematical model. The final solution is expressed in terms of the Weierstrass elliptic function. Periodic and solitary wave solutions of the RKL equation in the form of the traveling wave reduction are presented and illustrated.