Abstract
Publisher Summary This chapter summarizes some of the main results for the inverse scattering transform (IST) of the Nonlinear Schrodinger (NLS) equation with periodic boundary conditions. The nonlinear Schrodinger equation, scaled to represent physical units, is given by Yuen and Lake. The space NLS (sNLS) equation describes the space/time dynamics of the complex envelope function, ψ(x, t), of a deep-water wave train, which propagates in the +x direction as a function of time, t. The equation solves the Cauchy problem––that is, given the complex envelope at some initial time t = 0, ψ(x, 0), and evolve the dynamics for all space and time, ψ(x, t). The sea surface elevation, η(x, t), is computed from the complex envelope function, ψ(x, t). The chapter describes the “Time” NLS equation and its relation to physical experiments. A scaled form of the NLS equation is presented. The chapter describes the IST procedure that provides the analytic means for determining the long-time evolution of the solutions of the NLS equation.
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