TY - JOUR

T1 - An extension of the steepest descent method for Riemann-Hilbert problems

T2 - The small dispersion limit of the Korteweg-de Vries (KdV) equation

AU - Deift, P.

AU - Venakides, S.

AU - Zhou, X.

PY - 1998/1/20

Y1 - 1998/1/20

N2 - 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. USA 76, 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.

AB - 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. USA 76, 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.

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U2 - 10.1073/pnas.95.2.450

DO - 10.1073/pnas.95.2.450

M3 - Article

C2 - 11038618

AN - SCOPUS:0031881570

SN - 0027-8424

VL - 95

SP - 450

EP - 454

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

IS - 2

ER -