Designed for first-semester science students, this book focuses on building a foundational understanding of linear algebra, enhancing computational skills, and familiarizing readers with mathematical language. It offers precise definitions and helpful notations while presenting results clearly, though proofs are not included. Instead, the text is enriched with numerous examples and supportive arguments to aid comprehension, making mathematical literature more accessible.
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 periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way. Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations. The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation.
