Riemann-Hilbert Problem for the Small Dispersion Limit of the KdV Equation and Linear Overdetemined Systems of Euler-Poisson-Darboux Type

摘要

The Cauchy problem for the Korteweg-de Vries (KdV) equation (1.1) {u_t - 6uu_x + ∈~2u_(xxx) = 0, ∈ > 0, x, u ∈R, t ∈ R~+, u(x,t = 0, ∈) = U(x), in the zero-dispersion limit has been widely studied. The physical interest in this limit is due to the fact that it describes the phenomenon of shock waves in dissipation-less dispersive media. Dispersive shock waves are characterized by the appearance of rapid modulated oscillations. Gurevich and Pitaevskii [16] suggested that these oscillations could be modeled by the solution of the one-phase Whitham equations [30]. The multiphase or g-phase Whitham equations were derived by Flaschka, Forest, and McLaughlin [12]. Lax and Levermore [21] rigorously showed that the multiphase Whitham equations appear in the zero dispersion limit of the Cauchy problem for the KdV equation with asymptotically reflectionless initial data.