scholarly journals Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

2021 ◽  
Author(s):  
T. Buckmaster ◽  
P. Germain ◽  
Z. Hani ◽  
J. Shatah
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fourier Coefficients ◽  
Nonlinear Schrodinger Equation ◽  
Turbulence Theory ◽  
Wave Turbulence ◽  
Nonlinear Schrödinger ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Longtime Behavior

AbstractConsider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

scholarly journals On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 823
Author(s):  
Sergey Nazarenko ◽  
Avy Soffer ◽  
Minh-Binh Tran
Keyword(s):  
Porous Medium ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Porous Medium Equation ◽  
Nonlinear Schrodinger Equation ◽  
Turbulence Theory ◽  
Wave Turbulence ◽  
Random Potentials ◽  
Nonlinear Schrödinger

We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation.

scholarly journals The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities

2021 ◽  
Author(s):  
Rupert L. Frank ◽  
David Gontier ◽  
Mathieu Lewin
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Bound State ◽  
Nonlinear Schrodinger Equation ◽  
Best Constant ◽  
Nonlinear Schrödinger ◽  
The One ◽  
Bound State Case ◽  
Cubic Nonlinear Schrödinger Equation

AbstractIn this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator $$-\Delta +V(x)$$ - Δ + V ( x ) are raised to the power $$\kappa $$ κ is never given by the one-bound state case when $$\kappa >\max (0,2-d/2)$$ κ > max ( 0 , 2 - d / 2 ) in space dimension $$d\ge 1$$ d ≥ 1 . When in addition $$\kappa \ge 1$$ κ ≥ 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

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