Derivation of Rogue Waves in Integrable Systems

In this chapter, we derive general rogue waves in a wide variety of integrable systems, including those obtained in Chap. 1 but also many others, such as the Boussinesq equation, the Ablowitz-Ladik equation, the complex modified Korteweg-de Vries equation, the complex short pulse equation, the massive Thirring model, the Sasa-Satsuma equation, and the parity-time-symmetric NLS equation. We will use the bilinear method in all our derivations, since general rogue wave expressions out of this method are simpler and more explicit than those out of other methods. For the NLS equation, derivation of rogue waves by Darboux transformation is also presented, so that the reader can compare the two methods and their results. Dynamics of low-order rogue waves in these integrable systems are also graphically illustrated.