Asymptotics of semiclassical soliton ensembles: rigorous justification of the WKB approximation

Abstract

Rigorous pointwise asymptotics are established for semiclassical soliton ensembles (SSEs) of the focusing nonlinear Schrodinger equation using techniques of asymptotic analysis of matrix Riemann- Hilbert problems. The accumulation of poles in the eigenfunction is handled using a new method in which the residues are simultaneously interpolated at the poles by two distinct interpolants. The results justify the WKB approximation for the nonselfadjoint Zakharov-Shabat operator with real-analytic, bell-shaped, even potentials. The new technique introduced in this paper is applicable to other problems as well: (i) it can be used to provide a unified treatment by Riemann-Hilbert methods of the zero-dispersion limit of the Korteweg-de Vries equation with negative (soliton generating) initial as studied by Lax, Levermore, and Venakides, and (ii) it allows one to compute rigorous strong asymptotics for systems of discrete orthogonal polynomials. Many important problems in the theory of integrable systems and approximation theory can be recast as Riemann-Hilbert problems for a matrix-valued unknown. Via the connection with approximation theory, and specifically the theory of orthogonal polynomials, one can also study problems from the theory of random matrix ensembles and combinatorics. Roughly speaking, solving a Riemann-Hilbert problem amounts to reconstructing a sectionally meromorphic matrix from given homogeneous multiplicative at the boundary contours of the domains of meromorphy, from part data given at the prescribed singularities, and from a normalization condition. So, many asymptotic questions in integrable systems (e.g. long time behavior and singular perturbation theory) and approximation theory (e.g. behavior of orthogonal polynomials in the limit of large degree) amount to determining asymptotic properties of the solution matrix of a Riemann-Hilbert problem from given asymptotics of the jump conditions and principal part data. In recent years a collection of techniques has emerged for studying certain asymptotic problems of this sort. These techniques are analogous to familiar asymptotic methods for expanding oscillatory integrals, and we often refer to them as steepest-descent methods. The basic method first appeared in the work of Deift and Zhou (DZ93). The first applications were to Riemann-Hilbert problems without poles, in which the solution matrix is sectionally holomorphic. Later, some problems were studied in which there were a number of poles — a number held fixed in the limit of interest — in the solution matrix (see, for example, the paper (DKKZ96) on the long-time behavior of the Toda lattice with rarefaction initial data). The previous methods were extended to these more complicated problems through the device of making a local change of variable near each pole in some small domain containing the pole. The change of variable is chosen so that it has the effect of removing the pole at the cost of introducing an explicit jump on the boundary of the domain around the pole in which the transformation is made. The result is a Riemann-Hilbert problem for a sectionally holomorphic matrix, which can be solved asymptotically by pre-existing steepest-descent methods. Recovery of an approximation for the original sectionally meromorphic matrix unknown involves putting back the poles by reversing the explicit change of variables that was designed to get rid of them to begin with. Yet another category of Riemann-Hilbert problems consists of those problems where the number of poles is not fixed, but becomes large in the limit of interest, with the poles accumulating on some closed set F in the finite complex plane. A problem of this sort has been addressed (KMM00) by making an explicit transformation of the type described above in a single fixed domain G that contains the locus of accumulation F of all the poles. The transformation is chosen to get rid of all the poles at once. In order to specify it,