The inverse scattering transform for the KdV equation with step-like singular Miura initial profiles

DOI10.1063/1.4930001zbMath1328.35199arXiv1412.2184OpenAlexW3104009899MaRDI QIDQ3192686

Remling, Christian, Alexei Rybkin, Sergei M. Grudsky

Publication date: 13 October 2015

Published in: Journal of Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1412.2184

zbMATH Keywords

Hankel operatorKorteweg-de Vries equationTitchmarsh-Weyl function

Mathematics Subject Classification ID

Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) KdV equations (Korteweg-de Vries equations) (35Q53) Inverse problems for PDEs (35R30) Initial value problems for nonlinear higher-order PDEs (35G25) Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems (37K15)

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Cites Work

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• Geometry of KdV. IV: Abel sums, Jacobi variety, and theta function in the scattering case
• Global wellposedness of KdV in \(H^{-1}(\mathbb T,\mathbb R)\)
• 1-D Schrödinger operators with local point interactions on a discrete set
• Global well-posedness of Korteweg-de Vries equation in \(H^{-3/4}(\mathbb R)\)
• The Fredholm determinant method for the KdV equations
• Fredholm determinants and inverse scattering problems
• Solutions of the Korteweg-de Vries equation from irregular data
• Pole dynamics for elliptic solutions of the Korteweg-de Vries equation
• Introduction to the theory of Toeplitz operators with infinite index. Transl. from the Russian by Andrei Iacob
• Generalized reflection coefficients
• Meromorphic families of compact operators
• Spatial Analyticity of Solutions to Integrable Systems. I. The KdV Case
• Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials
• Sobolev mapping properties of the scattering transform for the Schr\"odinger equation
• The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line
• Exact Solution of the Korteweg—de Vries Equation for Multiple Collisions of Solitons
• Meromorphic solutions to the KdV equation with non-decaying initial data supported on a left half line
• On the numerical evaluation of Fredholm determinants
• Pole dynamics of interacting solitons and blowup of complex-valued solutions of KdV
• On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data: I. Schwartz-type perturbations
• Solutions to the korteweg - devries equation with irregular initial profile
• Long Time Asymptotics of the Korteweg-de Vries Equation
• Unique Continuation for the Korteweg–de Vries Equation
• Rational and elliptic solutions of the korteweg-de vries equation and a related many-body problem
• Strong regularizing effect of integrable systems
• Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
• Sharp global well-posedness for KdV and modified KdV on ℝ and 𝕋
• Soliton Theory and Hankel Operators
• On positive type initial profiles for the KdV equation
• An explicit characterization of Calogero–Moser systems