Explicit soliton asymptotics for the Korteweg–de Vries equation on the half-line

Abstract

There exists a distinctive class of physically significant evolution PDEs in one spatial dimension which can be treated analytically. A prototypical example of this class (which is referred to as integrable) is the Korteweg–de Vries equation. Integrable PDEs on the line can be analysed by the so-called inverse scattering transform (IST) method. A particularly powerful aspect of the IST is its ability to predict the large t behaviour of the solution. Namely, starting with initial data u(x, 0), IST implies that the solution u(x, t) asymptotes to a collection of solitons as t → ∞, x/t = O(1); moreover, the shapes and speeds of these solitons can be computed from u(x, 0) using only linear operations. One of the most important developments in this area has been the generalization of the IST from initial to initial-boundary value (IBV) problems formulated on the half-line. It can be shown that again u(x, t) asymptotes into a collection of solitons, where now the shapes and the speeds of these solitons depend both on u(x, 0) and on the given boundary conditions at x = 0. A major complication of IBV problems is that the computation of the shapes and speeds of the solitons involves the solution of a nonlinear Volterra integral equation. However, for a certain class of boundary conditions, called linearizable, this complication can be bypassed and the relevant computation is as effective as in the case of the problem on the line. Here, after reviewing the general theory for KdV, we analyse three different types of linearizable boundary conditions. For these cases, the initial conditions are (a) restrictions of one- and two-soliton solutions at t = 0; (b) profiles of certain exponential type and (c) box-shaped profiles. For each of these cases, by computing explicitly the shapes and the speeds of the asymptotic solitons, we elucidate the influence of the boundary.