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In this text the authors consider the Korteweg-de Vries (KdV) equation (u_t = - u_xxx + 6uu_x) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are...

This book is based on a course for first-semester science students, held by the second author at the University of Zurich several times. Its goal is threefold: to have students learn a minimal working knowledge of linear algebra, acquire some computational skills, and familiarize them with mathematical language to make mathematical literature more accessible. Therefore, we give precise definitions, introduce helpful notations, and state any results carefully worded. We provide no proofs of these results but typically illustrate them with numerous examples. Additionally, for better...

In this text the authors consider the Korteweg-de Vries (KdV) equation (u_t = - u_xxx + 6uu_x) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are...