A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation

DOI10.2307/2946540zbMath0771.35042OpenAlexW2062565740WikidataQ56386738 ScholiaQ56386738MaRDI QIDQ2366458

Publication date: 29 June 1993

Published in: Annals of Mathematics. Second Series (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.2307/2946540

zbMATH Keywords

inverse scattering method long-time behavior of nonlinear wave equations

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) KdV equations (Korteweg-de Vries equations) (35Q53) Riemann-Hilbert problems in context of PDEs (35Q15)

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PDEs on the Line, Riemann‐Hilbert Problems for the Shapes Formed by Bodies Dissolving, Melting, and Eroding in Fluid Flows, Asymptotics of matrix valued orthogonal polynomials on $[-1, 1$], Planar orthogonal polynomials as type I multiple orthogonal polynomials, Global conservative solutions of the nonlocal NLS equation beyond blow-up, Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces, Airy and Painlevé asymptotics for the mKdV equation, Three Lectures on “Fifty Years of KdV: An Integrable System”, Benjamin-Ono and Intermediate Long Wave Equations: Modeling, IST and PDE, Inverse Scattering and Global Well-Posedness in One and Two Space Dimensions, Dispersive Asymptotics for Linear and Integrable Equations by the ∂ ¯ $$\overline {\partial }$$ Steepest Descent Method, Asymptotics behavior for the integrable nonlinear Schrödinger equation with quartic terms: Cauchy problem, Thinning and conditioning of the circular unitary ensemble, Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation, A Riemann–Hilbert approach for the modified short pulse equation, Large time asymptotics for the cylindrical Korteweg–de Vries equation. I., Finite genus solutions to the Ablowitz‐Ladik equations, On the Long-Time Asymptotic Behavior of the Modified Korteweg--de Vries Equation with Step-like Initial Data, Superdiffusion in spin chains, Asymptotics of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane, Strong asymptotics of Jacobi-type kissing polynomials, Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary point, EXACT SOLUTION OF THE SIX-VERTEX MODEL WITH DOMAIN WALL BOUNDARY CONDITIONS: CRITICAL LINE BETWEEN DISORDERED AND ANTIFERROELECTRIC PHASES, Whitham equations and phase shifts for the Korteweg–de Vries equation, Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions, Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach, The asymptotic stability of solitons in the cubic NLS equation on the line, Mehler–Heine formulas for orthogonal polynomials with respect to the modified Jacobi weight, Decay and Scattering of Small Solutions of Pure Power NLS in ℝ with p > 3 and with a Potential, Integral equations for correlation functions of a quantum one-dimensional Bose gas, GLOBAL ASYMPTOTICS OF THE HAHN POLYNOMIALS, Magnetic properties of doped Heisenberg chains., The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line, Bessel-type asymptotic expansions via the Riemann–Hilbert approach, Random words, Toeplitz determinants and integrable systems. II, The eigenvalue problem for the focusing nonlinear Schrödinger equation: New solvable cases, Semiclassical Limit of Focusing NLS for a Family of Square Barrier Initial Data, A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials, Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on a finite interval, The generating function for the Bessel point process and a system of coupled Painlevé V equations, Double Spiral Singularities for a Flow of Regular Planar Curves, Introduction, Introduction, The massive Thirring system in the quarter plane, Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials, Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation, Semiclassical WKB problem for the non-self-adjoint Dirac operator with analytic potential, The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution, Connection formulas for the Ablowitz–Segur solutions of the inhomogeneous Painlevé II equation, The sine-Gordon equation in the semiclassical limit: Dynamics of fluxon condensates, The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$, Defocusing nonlocal nonlinear Schr\"odinger equation with step-like boundary conditions: long-time behavior for shifted initial data, High‐Order Soliton Solution of Landau–Lifshitz Equation, Transition asymptotics of Toeplitz determinants and emergence of Fisher–Hartwig representations, The Modified Korteweg-de Vries Equation on the Half-Line with a Sine-Wave as Dirichlet Datum, Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation, The modified Korteweg-de Vries equation on the quarter plane with t-periodic data, The Riemann-Hilbert Approach to Initial-Boundary Value Problems for Integrable Coherently Coupled Nonlinear Schrodinger Systems on the Half-Line, On the origins of Riemann–Hilbert problems in mathematics*, Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems, Singular asymptotics for solutions of the inhomogeneous Painlevé II equation, Application of the Nonlinear Steepest Descent Method to the Coupled Sasa-Satsuma Equation, Long-time Asymptotics for the Integrable Discrete Nonlinear Schrödinger Equation: the Focusing Case, The multiplicative constant for the Meijer-G kernel determinant, The local universality of Muttalib–Borodin ensembles when the parameter θ is the reciprocal of an integer, A Riemann-Hilbert approach to the initial-boundary value problem for Kundu-Eckhaus equation on the half line, Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems, Riemann–Hilbert problem to the modified Korteveg–de Vries equation: Long-time dynamics of the steplike initial data, Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit, Inverse Scattering Transform for the Defocusing Manakov System with Nonzero Boundary Conditions, Asymptotics of a cubic sine kernel determinant, Soliton Theory and Hankel Operators, Gap probabilities in the bulk of the Airy process, Inverse Scattering Transform for the Defocusing Manakov System with Non-Parallel Boundary Conditions at Infinity, Long-time asymptotics for the complex nonlinear transverse oscillation equation, From Painlevé to Zakharov–Shabat and beyond: Fredholm determinants and integro-differential hierarchies, A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, Asymptotic behavior of dispersive waves in a spiral structure at large times, Asymptotic stability of solitons for mKdV, Global rigidity and exponential moments for soft and hard edge point processes, Emptiness formation probability of the six-vertex model and the sixth Painlevé equation, A vector Geng-Li model: new nonlinear phenomena and breathers on periodic background waves, Riemann boundary value problems for monogenic functions on the hyperplane, Soliton resolution for the complex short pulse equation with weighted Sobolev initial data in space-time solitonic regions, Long time and Painlevé-type asymptotics for the Sasa-Satsuma equation in solitonic space time regions, Soliton resolution for the Wadati-Konno-Ichikawa equation with weighted Sobolev initial data, Universality of a double scaling limit near singular edge points in random matrix models, Darboux transformation and solitonic solution to the coupled complex short pulse equation, Global asymptotics of Krawtchouk polynomials -- a Riemann-Hilbert approach, Type II Hermite-Padé approximation to the exponential function, Long-time asymptotics for the modified complex short pulse equation, The initial-boundary value problems of the new two-component generalized Sasa-Satsuma equation with a $4\times 4$ matrix Lax pair, Painlevé-type asymptotics of an extended modified KdV equation in transition regions, Soliton resolution for the short-pulse equation, Riemann-Hilbert approach and long-time asymptotics of the positive flow short-pulse equation, Defocusing NLS equation with nonzero background: large-time asymptotics in a solitonless region, Riemann-Hilbert problem and matrix biorthogonal polynomials, The short pulse equation by a Riemann-Hilbert approach, Orthogonal polynomials for a class of measures with discrete rotational symmetries in the complex plane, N-soliton solutions for the coupled extended modified KdV equations via Riemann-Hilbert approach, Long-time asymptotics for the integrable nonlocal focusing nonlinear Schrödinger equation for a family of step-like initial data, Long-time asymptotics for the spin-1 Gross-Pitaevskii equation, Exotic localized vector waves in a two-component nonlinear wave system, Riemann-Hilbert problem for a fourth-order dispersive nonlinear Schrödinger equation on the half-line, Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions in the presence of a discrete spectrum, Soliton and breather solutions for a fifth-order modified KdV equation with a nonzero background, Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy, Lower tail of the KPZ equation, Long-time asymptotic behavior for the discrete defocusing mKdV equation, On the evolution of scattering data under perturbations of the Toda lattice, The largest real eigenvalue in the real Ginibre ensemble and its relation to the Zakharov-Shabat system, Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann-Hilbert approach, Long-time asymptotics for a mixed nonlinear Schrödinger equation with the Schwartz initial data, Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent, On the Riemann-Hilbert problem of the Kundu equation, Fokas-Lenells equation: three types of Darboux transformation and multi-soliton solutions, Long-time asymptotics of a three-component coupled nonlinear Schrödinger system, A Riemann-Hilbert approach to a generalized nonlinear Schrödinger equation on the quarter plane, Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach, Long-time asymptotics for the Degasperis-Procesi equation on the half-line, Gap probability of the circular unitary ensemble with a Fisher-Hartwig singularity and the coupled Painlevé V system, Long-time large-distance asymptotics of the transverse correlation functions of the XX chain in the spacelike regime, Long-time asymptotics of the coupled modified Korteweg-de Vries equation, The generalized Dirichlet to Neumann map for the KdV equation on the half-line, Inverse scattering transform for the Gerdjikov-Ivanov equation with nonzero boundary conditions, Long-time asymptotic behavior for the complex short pulse equation, Long-time asymptotics for the nonlocal nonlinear Schrödinger equation with step-like initial data, The Ostrovsky-Vakhnenko equation: A Riemann-Hilbert approach, Conserved energies for the one dimensional Gross-Pitaevskii equation, Finite gap integration of the derivative nonlinear Schrödinger equation: a Riemann-Hilbert method, Numerical inverse scattering for the sine-Gordon equation, The matching condition for larger size Riemann-Hilbert problems, Asymptotics of a Fredholm determinant corresponding to the first bulk critical universality class in random matrix models, $N$-soliton solutions for the nonlocal Fokas-Lenells equation via RHP, On the Riemann-Hilbert problem for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation, Mixed type Hermite-Padé approximants for a Nikishin system, Matrix biorthogonal polynomials: eigenvalue problems and non-Abelian discrete Painlevé equations. A Riemann-Hilbert problem perspective, Integrable nonlinear evolution equations on a finite interval, The nonlinear steepest descent approach for long time behavior of the two-component coupled Sasa-Satsuma equation with a $5 \times 5$ Lax pair, Soliton resolution for the focusing modified KdV equation, Construction of solutions and asymptotics for the defocusing NLS with periodic boundary data, Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum, Long-time asymptotic behavior for a fourth-order dispersive nonlinear Schrödinger equation, Asymptotics of recurrence relation coefficients, Hankel determinant ratios, and root products associated with Laurent polynomials orthogonal with respect to varying exponential weights, Coupled derivative nonlinear Schrödinger III equation: Darboux transformation and higher-order rogue waves in a two-mode nonlinear fiber, Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schrödinger model, Homogenized spectral problems for exactly solvable operators: asymptotics of polynomial eigenfunctions, The $k$-tacnode process, Gaussian unitary ensembles with pole singularities near the soft edge and a system of coupled Painlevé XXXIV equations, A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle, Interactions and asymptotics of dispersive shock waves -- Korteweg-de Vries equation, Laguerre polynomials and transitional asymptotics of the modified Korteweg-de Vries equation for step-like initial data, A priori bounds for KdV equation below $H^{- \frac{3}{4}}$, Phase transitions and equilibrium measures in random matrix models, A matrix model with a singular weight and Painlevé III, Modified KdV equation with higher order dispersion terms, Evolution of initial discontinuity for the defocusing complex modified KdV equation, Asymptotics of polynomials orthogonal on a cross with a Jacobi-type weight, Asymptotics for the Sasa-Satsuma equation in terms of a modified Painlevé II transcendent, The unified transform method to initial-boundary value problem for a coupled cubic-quintic nonlinear Schrödinger system, Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation, Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, Gaussian fluctuations and moderate deviations of eigenvalues in unitary invariant ensembles, Self-similar asymptotics for solutions to the intermediate long-wave equation, Riemann-Hilbert approach for an initial-boundary value problem of the two-component modified Korteweg-de Vries equation on the half-line, Large $n$ limit of Gaussian random matrices with external source. II, Asymptotics for Laguerre polynomials with large order and parameters, Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: without solitons, On an isomonodromy deformation equation without the Painlevé property, Riemann-Hilbert problem for Camassa-Holm equation with step-like initial data, Universality conjecture and results for a model of several coupled positive-definite matrices, On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential. I, On critical behaviour in systems of Hamiltonian partial differential equations, Tronquée solutions of the Painlevé equation PI, On the inverse scattering method for integrable PDEs on a star graph, Bounds for KdV and the 1-d cubic NLS equation in rough function spaces