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. 117â€“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 PainlevĂ©-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 -