An Extension of the Steepest Descent Method for Riemann-Hilbert Problems: the Small Dispersion Limit of the Korteweg-de Vries (kdv) Equation

This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way. We present, in particular, an algorithm, to obtain the support of the Riemann-Hilbert problem for leading asymptotics. Applying this extended method to small dispersion KdV (Korteweg-de Vries) equation, we (i) recover the variational formulation of P. D. Lax and C. D. Levermore [(1979) Proc. Natl. Acad. Sci. USA76, 3602-3606] for the weak limit of the solution, (ii) derive, without using an ansatz, the hyperelliptic asymptotic solution of S. Venakides that describes the oscillations; and (iii) are now able to compute the phase shifts, integrating the modulation equations exactly. The procedure of this paper is a version of fully nonlinear geometrical optics for integrable systems. With some additional analysis the theory can provide rigorous error estimates between the solution and its computed asymptotic expression.