Dispersive equations and nonlinear waves. Generalized Korteweg-de Vries, nonlinear Schrödinger, wave and Schrödinger maps

DOI10.1007/978-3-0348-0736-4zbMath1304.35003OpenAlexW2485812420MaRDI QIDQ390782

Daniel Tataru, Herbert Koch, Monica Visan

Publication date: 8 January 2014

Published in: Oberwolfach Seminars (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/978-3-0348-0736-4

zbMATH Keywords

Fourier transformexistenceRiemannian manifoldwell-posednessRiemannian manifoldsSchrödinger equationStrichartz estimatenonlinear waveKadomtsev-Petviashvili equationdispersive equationblowup-solutiongeometric evolutionsharmonic map equationlarge data casePaley-Smale theorySchrödinger propagatorsmall data casewave map equation

Mathematics Subject Classification ID

KdV equations (Korteweg-de Vries equations) (35Q53) NLS equations (nonlinear Schrödinger equations) (35Q55) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Research exposition (monographs, survey articles) pertaining to partial differential equations (35-02) Time-dependent Schrödinger equations and Dirac equations (35Q41) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02)

Related Items

Strichartz estimates for \(N\)-body Schrödinger operators with small potential interactions ⋮ Charge conjugation approach to scattering for the Hartree type Dirac equations with chirality ⋮ Infinite Time Blow-Up Solutions to the Energy Critical Wave Maps Equation ⋮ The scattering map determines the nonlinearity ⋮ Small data scattering of semirelativistic Hartree equation ⋮ Optical solitons and periodic solutions of the \((2 + 1)\)-dimensional nonlinear Schrödinger's equation ⋮ The radial mass-subcritical NLS in negative order Sobolev spaces ⋮ On small data scattering of Hartree equations with short-range interaction ⋮ Small data well-posedness for derivative nonlinear Schrödinger equations ⋮ Quantitative derivation and scattering of the 3D cubic NLS in the energy space ⋮ The unconditional uniqueness for the energy-supercritical NLS ⋮ CONDITIONAL LARGE INITIAL DATA SCATTERING RESULTS FOR THE DIRAC–KLEIN–GORDON SYSTEM ⋮ Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation ⋮ Random data theory for the cubic fourth-order nonlinear Schrödinger equation ⋮ Equivariant Schrödinger map flow on two dimensional hyperbolic space ⋮ Large data mass-subcritical NLS: critical weighted bounds imply scattering ⋮ Scattering results for Dirac Hartree-type equations with small initial data ⋮ Large data well-posedness in the energy space of the Chern-Simons-Schrödinger system ⋮ Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: energy critical case ⋮ Dispersive estimates for nonhomogeneous radial phases: an application to weakly dispersive equations and water wave models ⋮ Scattering in \(H^{1}\) for the intercritical NLS with an inverse-square potential ⋮ Recovery of a spatially-dependent coefficient from the NLS scattering map ⋮ Stability estimates for the recovery of the nonlinearity from scattering data ⋮ Global well-posedness for the fifth order Kadomtsev-Petviashvili II equation in three dimensional space ⋮ Global well-posedness for the defocusing Hartree equation with radial data in ℝ4 ⋮ Global well-posedness, scattering and blow-up for the generalized Boussinesq equation in high dimensions below the ground state energy ⋮ Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases ⋮ Asymptotic behavior of incompressible Schrödinger flow for small data in three dimensions ⋮ Optimal bilinear control of stochastic nonlinear Schrödinger equations: mass-(sub)critical case ⋮ Global well-posedness for the Sobolev critical nonlinear Schrödinger system in four space dimensions ⋮ Global Well-Posedness and Scattering for Mass-Critical, Defocusing, Infinite Dimensional Vector-Valued Resonant Nonlinear Schrödinger System ⋮ Almost sure well-posedness and scattering of the 3D cubic nonlinear Schrödinger equation ⋮ On Scattering for the Defocusing Quintic Nonlinear Schrödinger Equation on the Two-Dimensional Cylinder ⋮ Stable self-similar blow-up dynamics for slightly \(L^2\)-supercritical generalized KdV equations ⋮ Numerical simulations for the energy-supercritical nonlinear wave equation ⋮ Small Data Scattering for Cubic Dirac Equation with Hartree Type Nonlinearity in $\mathbb{R}^{1+3}$ ⋮ Local wellposedness for the critical nonlinear Schrödinger equation on \( \mathbb{T}^3 \) ⋮ On the division problem for the wave maps equation ⋮ Small data scattering for semi-relativistic equations with Hartree type nonlinearity ⋮ Small data scattering of 2d Hartree type Dirac equations ⋮ Scattering theory for the radial \(\dot{H}^{\frac{1}{2}}\)-critical wave equation with a cubic convolution ⋮ Stability of the multi-solitons of the modified Korteweg–de Vries equation ^* ⋮ Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on \(\mathbb{T}^3\) ⋮ Transference of bilinear restriction estimates to quadratic variation norms and the Dirac-Klein-Gordon system ⋮ The defocusing quintic NLS in four space dimensions ⋮ The energy-critical NLS with inverse-square potential ⋮ Stable self-similar blow-up dynamics for slightly \(L^2\)-supercritical generalized KdV equations ⋮ Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on \({\mathbb{R}^3}\) ⋮ On global well-posedness and scattering for the massive Dirac-Klein-Gordon system ⋮ Local existence and uniqueness of Navier-Stokes-Schrödinger system ⋮ Unnamed Item ⋮ Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: high dimensions ⋮ The Strichartz estimates for the damped wave equation and the behavior of solutions for the energy critical nonlinear equation ⋮ Global dynamics of the generalized fifth-order KdV equation with quintic nonlinearity ⋮ Long time behavior of solutions to the mKdV ⋮ Scattering for the non-radial energy-critical inhomogeneous NLS ⋮ Transverse stability of line soliton and characterization of ground state for wave guide Schrödinger equations ⋮ The global well-posedness and scattering for the 5-dimensional defocusing conformal invariant NLW with radial initial data in a critical Besov space ⋮ Global well-posedness and scattering for the defocusing \(\dot{H}^{1/2}\)-critical nonlinear Schrödinger equation in \(\mathbb{R}^2\) ⋮ Unconditional well-posedness for the energy-critical nonlinear damped wave equation ⋮ Low regularity conservation laws for integrable PDE ⋮ Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations ⋮ Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces ⋮ Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on ⋮ Wellposedness of NLS in modulation spaces ⋮ Near resonant approximation of the rotating stratified Boussinesq system on a 3-torus ⋮ Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations