Physical Derivation of Integrable Nonlinear Wave Equations

In this chapter, we derive many integrable systems as governing equations for diverse physical processes. Examples include the nonlinear Schrödinger equation for wave packet propagation in deep water, optical fibers, and unmagnetized plasma, the derivative nonlinear Schrödinger equation for nonlinear Alfvén waves in magnetized plasma, Manakov equations for light transmission in randomly-birefringent optical fibers, the long-wave-short-wave interaction model in water of finite depth, the three-wave resonant interaction system in water waves and optics, and Davey-Stewartson equations for two-dimensional wave packets in water of finite depth. The primary method of our derivation is the multi-scale perturbation method. These physical backgrounds of the underlying integrable systems will not only motivate our mathematical studies of rogue waves in these integrable systems, but also link our mathematical rogue solutions to physical experiments which we will also describe in later chapters.