Solitons, numerical experiments, and that mysterious lady

The interesting article by Thierry Dauxois on “Fermi, Pasta, Ulam, and a Mysterious Lady” (Physics Today, Physics Today 0031-9228 61 1 2008 55  https://doi.org/10.1063/1.2835154 January 2008, page 55 ) relates the subject of solitons to that of the Fermi-Pasta-Ulam (FPU) problem. The term “soliton” was introduced by Norman Zabusky and Martin Kruskal ¹ in 1965 because the nonlinear waves studied did not lose their identity after colliding. In a sense, they resembled particles. The study by Zabusky and Kruskal was a numerical one of the Korteweg–de Vries equation, but the motivation was to study the propagation of waves in a collisionless plasma containing a magnetic field. Fifty years ago John Adlam and I studied that problem ² and found an analytical solution for strong, collision-free hydromagnetic solitary waves for Alfvén Mach numbers less than 2. The solution was not valid for faster, stronger waves. Further work in 1960 dealt with the excitation of a train of such waves; ³ that time the equations were solved numerically. The work with Adlam seems to have been largely overlooked until recently, ⁴ presumably because it predated the term “soliton.”