Numerical construction of nonlinear wave-train solutions of the periodic Korteweg–de Vries equation

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Abstract
I discuss a general approach for the numerical construction of exact, nonlinear wave-train solutions to the periodic Korteweg–de Vries (KdV) equation. The method is based upon the periodic inverse scattering transform (IST), a nonlinear generalization of ordinary Fourier series. In this approach, the solution to the KdV equation is represented by a linear superposition of nonlinearly interacting ‘‘hyperelliptic functions’’ which are the nonlinear ‘‘oscillation modes’’ or ‘‘degrees of freedom’’ of the equation; the amplitudes of the nonlinear modes are the constants of the motion for KdV evolution. Using the periodic IST formulation, I numerically construct several low-degree-of-freedom wave trains and discuss some of their physical properties. The approach given here depends explicitly on the application of methods from the field of algebraic geometry. Most of the examples presented are solutions to the KdV equation which have not been previously considered; the solutions are ‘‘complex’’ in the sense that, instead of being a single cnoidal wave, they are ‘‘multicnoidal’’ or ‘‘polycnoidal.’’ The IST spectrum often provides a much simpler interpretation of the wave motion than that given by the linear Fourier transform. This occurs primarily because the nonlinear wave trains constructed herein have a small number of IST modes; on the other hand, these wave trains generally require a large number of linear Fourier modes for their description.