Conserved energies for the cubic nonlinear Schrödinger equation in one dimension

2016/07/31 by Herbert Koch, Daniel Tataru
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Conserved quantity #Dimension (graph theory) #Korteweg–de Vries equation #Law #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Schrödinger equation #Nonlinear Waves and Solitons #Nonlinear system #Norm (philosophy) #Physics #Pure mathematics #Quantum mechanics #Schrödinger equation #math.AP #msc:35Q55 #msc:37K10

paper · pdf · doi:10.1215/00127094-2018-0033

published as Duke Math. J. 167, no. 17 (2018), 3207-3313 · 72 pages, small corrections

arxiv created 2018/05/23 · openalex publication_date 2018/10/26 · arxiv updated 2018/11/14 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/06

Abstract

We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension. We prove that for each s>−12 there exists a conserved energy which is equivalent to the Hs-norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every s≥−1.

Citations

• The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems
• A-priori bounds for KdV equation below H^-3/4
• Well-posedness and scattering for the KP-II equation in a critical space
• A Steepest Descent Method for Oscillatory Riemann--Hilbert Problems. Asymptotics for the MKdV Equation
• Combinatorics on words
• Energy and local energy bounds for the 1-D cubic NLS equation in H^-1/4