An extension of the steepest descent method for Riemann-Hilbert problems: The small dispersion limit of the Korteweg-de Vries (KdV) equation | Zendy

Percy Deift | Zendy; Stephanos Venakides | Zendy; Xin Zhou | Zendy

Open Access

An extension of the steepest descent method for Riemann-Hilbert problems: The small dispersion limit of the Korteweg-de Vries (KdV) equation

Author(s) -

Percy Deift,

Stephanos Venakides,

Xin Zhou

Publication year - 1998

Publication title -

proceedings of the national academy of sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 5.011

H-Index - 771

eISSN - 1091-6490

pISSN - 0027-8424

DOI - 10.1073/pnas.95.2.450

Subject(s) - method of steepest descent , integrable system , korteweg–de vries equation , mathematics , ansatz , lax pair , limit (mathematics) , gradient descent , duality (order theory) , riemann hypothesis , mathematical analysis , mathematical physics , convergence (economics) , dispersion (optics) , nonlinear system , physics , pure mathematics , quantum mechanics , machine learning , artificial neural network , computer science , economics , economic growth

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.

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