01547nas a2200133   4500008004100000245008700041210006900128260000900197520111200206100002001318700001801338700002101356856003601377       2011                        en  d00aNumerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations0 aNumerical Study of breakup in generalized Kortewegde Vries and K  bSIAM3 aThis article is concerned with a conjecture in [B. Dubrovin, Comm. Math. Phys., 267 (2006), pp. 117–139] on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasi-linear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behavior of their solution near the point of gradient catastrophe for the transport equation is described by a special solution of a Painlevé-type equation. This local description holds also for solutions to equations where blowup can occur in finite time. Furthermore, it is shown that a solution of the dispersive equations away from the point of gradient catastrophe is approximated by a solution of the transport equation with the same initial data, modulo terms of order $\\\\epsilon^2$, where $\\\\epsilon^2$ is the small dispersion parameter. Corrections up to order $\\\\epsilon^4$ are obtained and tested numerically.1 aDubrovin, Boris1 aGrava, Tamara1 aKlein, Christian  uhttp://hdl.handle.net/1963/4951