Abstract. We apply the method of nonlinear steepest descent to compute the longtime asymptotics of the Camassa-Holm equation for decaying initial data, completing previous results by A. Boutet de Monvel and D. Shepelsky.

An initial boundary-value problem for the modified Korteweg-de Vries equation on the halfline, 0 < x < ∞, t > 0, is analysed by expressing the solution q(x, t) in terms of the solution of a matrix Riemann-Hilbert (RH) problem in the complex k-plane. This RH problem has explicit (x, t) dependence and it involves certain functions of k referred to as the spectral functions. Some of these functions are defined in terms of the initial condition q(x, 0) = q 0 (x), while the remaining spectral functions are defined in terms of the boundary values q(0, t) = g 0 (t), qx(0, t) = g 1 (t), and qxx(0, t) = g 2 (t). The spectral functions satisfy an algebraic global relation which characterizes, say, g 2 (t) in terms of {q 0 (x), g 0 (t), g 1 (t)}. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

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