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## A steepest descent method for oscillatory Riemann–Hilbert problems: asymptotics for the MKdV equation (1993)

Venue: | Ann. of Math |

Citations: | 303 - 27 self |

### Citations

96 |
Scattering and inverse scattering for first order systems,
- Beals, Coifman
- 1984
(Show Context)
Citation Context ...he KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, etc., and also to “integrable” ordinary differential equations such as the Painlevé transcendents. As described, for example, in [IN] or =-=[BC]-=-, the inverse scattering method for the MKdV equation leads to a Riemann-Hilbert factorization problem for a 2 × 2 matrix valued function m = m(·; x, t) analytic in C\R, (1) where m+(z) = m−(z)vx,t, z... |

76 |
The Isomonodromic Deformation Method in the Theory of Painlevé Equations
- Novokshenov
- 1986
(Show Context)
Citation Context ...uch as the KdV, nonlinear Schrödinger (NLS), and Boussinesq equations, etc., and also to “integrable” ordinary differential equations such as the Painlevé transcendents. As described, for example, in =-=[IN]-=- or [BC], the inverse scattering method for the MKdV equation leads to a Riemann-Hilbert factorization problem for a 2 × 2 matrix valued function m = m(·; x, t) analytic in C\R, (1) where m+(z) = m−(z... |

18 |
Asymptotic behavior of nonlinear wave systems integrated by the inverse method
- Zakharov, Manakov
- 1976
(Show Context)
Citation Context ...r wave equations solvable by the inverse scattering method, was first carried out by Manakov [M] and by Ablowitz and Newell [AN] in 1973. The decisive step was taken in 1976 when Zakharov and Manakov =-=[ZM]-=- were able to write down precise formulae, depending explicitly on the initial data, for the leading asymptotics for the KdV, NLS, and sine-Gordon equations, in the physically interesting region x = O... |

8 |
Asymptotics of solutions of the nonlinear Schrödinger equation and isomonodromic deformations of systems of linear differential equations
- Its
- 1981
(Show Context)
Citation Context ...in [BS] and in [N], involves an ansatz for the asymptotic form of the solution and utilizes techniques that are somewhat removed from the classical framework of Riemann-Hilbert problems. In 1981, Its =-=[I]-=- returned to a method first proposed in 1973 by Manakov in [M], which was tied more closely to standard methods for the inverse problem. In [I] the Riemann-Hilbert problem was conjugated, up to small ... |

7 |
The decay of the continuous spectrum for solutions of the Korteweg-de Vries equation
- ABLOWITZ, NEWELL
- 1973
(Show Context)
Citation Context ...)x,t, w± = ±(b± − I). Significant work on the long-time behavior of nonlinear wave equations solvable by the inverse scattering method, was first carried out by Manakov [M] and by Ablowitz and Newell =-=[AN]-=- in 1973. The decisive step was taken in 1976 when Zakharov and Manakov [ZM] were able to write down precise formulae, depending explicitly on the initial data, for the leading asymptotics for the KdV... |

7 | Asymptotic behavior of solutions of the Korteweg-de Vries equation - Buslaev, Sukhanov - 1905 |

5 |
Asymptotic solutions of the Korteweg de Vries equation
- Segur, Ablowitz
- 1977
(Show Context)
Citation Context ...n x = O(t). A complete description of the leading asymptotics of the solution of the Cauchy problem, with connection formulae between different asymptotic regions, was presented by Ablowitz and Segur =-=[AS]-=-, but without precise information on the phase. The asymptotic formulae of Zakharov and Manakov were rigorously justified and extended to all orders by Buslaev and Sukhanov [BS 1–2] in the case of the... |

4 |
Use of the determinant representation of solutions of the Korteweg–de Vries equation for the investigation of their asymptotic behavior for large times, Uspekhi Mat. Nauk 36:4
- Buslaev
- 1981
(Show Context)
Citation Context ...ations. This technique provides a viable, and in principle, rigorous approach to the question of long-time asymptotics for a wide class of nonlinear wave equations (see [IN]). Finally we note that in =-=[B]-=-, Buslaev derived asymptotic formulae for the KdV equation from an exact determinant formula for the solution of the inverse problem. In our approach we consider the Riemann-Hilbert problem (1) direct... |

4 |
Nonlinear Fraunhofer diffraction
- Manakov
- 1974
(Show Context)
Citation Context ...i 21 where wx,t = (w+)x,t + (w−)x,t, w± = ±(b± − I). Significant work on the long-time behavior of nonlinear wave equations solvable by the inverse scattering method, was first carried out by Manakov =-=[M]-=- and by Ablowitz and Newell [AN] in 1973. The decisive step was taken in 1976 when Zakharov and Manakov [ZM] were able to write down precise formulae, depending explicitly on the initial data, for the... |

2 |
Asymptotics as t → ∞ of solution of Cauchy problem for nonlinear Schrödinger equation., Dokl
- Novokshenov
(Show Context)
Citation Context ...on the phase. The asymptotic formulae of Zakharov and Manakov were rigorously justified and extended to all orders by Buslaev and Sukhanov [BS 1–2] in the case of the KdV equation, and by Novokshenov =-=[N]-=- in the case of NLS. The method of Zakharov and Manakov, pursued rigorously in [BS] and in [N], involves an ansatz for the asymptotic form of the solution and utilizes techniques that are somewhat rem... |