TY - JOUR T1 - Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations JF - SIAM J. Appl. Math. 71 (2011) 983-1008 Y1 - 2011 A1 - Boris Dubrovin A1 - Tamara Grava A1 - Christian Klein AB - This 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. PB - SIAM UR - http://hdl.handle.net/1963/4951 U1 - 4732 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER -