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Integrable PDEs on the line can be analyzed by the so-called InverseScattering Transform (IST) method. A particularly powerful aspect of the IST isits ability to predict the large $t$ behavior of the solution. Namely, startingwith initial data $u(x,0)$, IST implies that the solution $u(x,t)$ asymptotesto a collection of solitons as $t \to \infty$, $x/t = O(1)$; moreover theshapes and speeds of these solitons can be computed from $u(x,0)$ using only{\it linear} operations. One of the most important developments in this areahas 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 speedsof these solitons depend both on $u(x,0)$ and on the given boundary conditionsat $x = 0$. A major complication of IBV problems is that the computation of theshapes and speeds of the solitons involves the solution of a {\it nonlinear}Volterra integral equation. However, for a certain class of boundaryconditions, called linearizable, this complication can be bypassed and therelevant computation is as effective as in the case of the problem on the line.Here, after reviewing the general theory for KdV, we analyze three differenttypes of linearizable boundary conditions. For these cases, the initialconditions are: (a) restrictions of one and two soliton solutions at $t = 0$;(b) profiles of certain exponential type; (c) box-shaped profiles. For each ofthese cases, by computing explicitly the shapes and the speeds of theasymptotic solitons, we elucidate the influence of the boundary.

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