On the modified fractional Korteweg–de Vries and related equations

Abstract We consider in this paper modified fractional Korteweg–de Vries and related equations (modified Burgers–Hilbert Whitham). They have the advantage with respect to usual KdV equation a defocusing case different dynamics. will distinguish weakly dispersive where phase velocity is unbounded for low frequencies tends zero at infinity strongly vanishes origin goes infinity. In former case, nonlinear hyperbolic effects dominate large data, leading possibility of shock formation though manifest small initial data scattering possible. latter finite time blow-up possible focusing but not formation. global existence expected energy subcritical while supercritical case. establish rigorously shocks location being explicitly computed most results on are derived via numerical simulations, solutions. Moreover, result can be extended some generalized nonlinearity. also comment briefly BBM versions those equations.