An undular bore solution for the higher-order Korteweg–de Vries equation

Undular bores describe the evolution and smoothing out of an initial step in mean height and are frequently observed in both oceanographic and meteorological applications. The undular bore solution for the higher-order Korteweg–de Vries (KdV) equation is derived, using an asymptotic transformation which relates the KdV equation and its higher-order counterpart. The higher-order KdV equation considered includes all possible third-order correction terms (where the KdV equation retains second-order terms). The asymptotic transformation is then applied to the KdV undular bore solution to obtain the higher-order undular bore. Examples of higher-order undular bores, describing both surface and internal waves, are presented. Key properties, such as the amplitude and speed of the lead soliton and the width of the bore, are found. An excellent comparison is obtained between the analytical and numerical solutions. Also, it is illustrated how an asymptotic transformation and numerical solutions can be combined to generate hybrid asymptotic-numerical solutions, thus avoiding the severe instabilities associated with numerical schemes for the higher-order KdV equation.