From weak discontinuities to nondissipative shock waves

An analysis is presented of the effect of weak dispersion on transitions from weak to strong discontinuities in inviscid fluid dynamics. In the neighborhoods of transition points, this effect is described by simultaneous solutions to the Korteweg—de Vries equation ut′ + uux″ + uxxx‴ = 0 and fifth-order nonautonomous ordinary differential equations. As x2 + t2 →∞, the asymptotic behavior of these simultaneous solutions in the zone of undamped oscillations is given by quasi-simple wave solutions to Whitham equations of the form ri(t, x) = tli x/t2.