ON THE LONG-TIME ASYMPTOTIC BEHAVIOR OF THE MODIFIED KORTEWEG-DE VRIES EQUATION WITH STEP-LIKE INITIAL DATA
Abstract
We study the long-time asymptotic behavior of the solution q(x, t), x is an element of R, t is an element of R+, of the modified Korteweg-de Vries equation (MKdV) q(t) + 6q(2)q(x) + q(xxx) = 0 with step-like initial datum q(x,0) -> {( c- for x -> -infinity,)(c+ for x -> +infinity,) with c(-) > c(+) >= 0. For the step initial data q(x, 0) = {(c+) (c- for x 0) the solution develops an oscillatory region called the dispersive shock wave region that connects the two constant regions c(+) and c(-).
We show that the dispersive shock wave is described by a modulated periodic traveling wave solution of the MKdV equation where the modulation parameters evolve according to a Whitham modulation equation. The oscillatory region is expanding within a cone in the (x,t) plane defined as -6c(-)(2) + 12c(+)(2) + > 1.
For step-like initial data we show that the solution decomposes for long times into three main regions: (1) a region where solitons and breathers travel with positive velocities on a constant background c(+); (2) an expanding oscillatory region (that generically contains breathers); (3) a region of breathers traveling with negative velocities on the constant background c(-). When the oscillatory region does not contain breathers, the form of the asymptotic solution coincides up to a phase shift with the dispersive shock wave solution obtained for the step initial data.
The phase shift depends on the solitons, the breathers, and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers, and radiation.