The orbital stability of the cnoidal waves of the Korteweg–de Vries equation

The cnoidal wave solution of the integrable Korteweg–de Vries equation is the most basic of its periodic solutions. Following earlier work where the linear stability of these solutions was established, we prove in this Letter that cnoidal waves are (nonlinearly) orbitally stable with respect to so-called subharmonic perturbations: perturbations that are periodic with period any integer multiple of the cnoidal-wave period. Our method of proof combines the construction of an appropriate Lyapunov function with the seminal results of Grillakis, Shatah and Strauss (1987, 1990) [17] and [18]. The integrability of the Korteweg–de Vries equation is used in that we need the presence of at least one extra conserved quantity in addition to those expected from the Lie point symmetries of the equation.