Abstract.
We consider the small-dispersion and small-diffusion nonlinear Schrödinger equation \( -i \dot{u} = -\delta_{1} \Delta u - i \delta _{2} \Delta u + \mid u \mid ^{2}u + \zeta _{\omega} (t, x)\), \( 1 \geq \delta : = \sqrt{\delta^{2}_{1} + \delta^{2}_{2}} > 0\), where the space-variable x belongs to the unit n-cube (\( n \leq 3 \)) and u satisfies Dirichlet boundary conditions. Assuming that the force \( \zeta \) is a zero-meanvalue random field, smooth in x and stationary in t with decaying correlations, we prove that the C m-norms in x with \( m \geq 3 \) of solutions u, averaged in ensemble and locally averaged in time, are larger than \( \delta ^{-\kappa m} \), \( \kappa \approx 1/5 \). This means that the length-scale of a solution u decays with \( \delta \) as its positive degree (at least, as \( \delta^{\kappa} \) and - in a sense - proves existence of turbulence for this equation.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Submitted: August 1996, revised version: May 1997
Rights and permissions
About this article
Cite this article
Kuksin, S. On Turbulence in Nonlinear Schrödinger Equations. Geom. Funct. Anal. 7, 783–822 (1997). https://doi.org/10.1007/s000390050026
Published:
Issue Date:
DOI: https://doi.org/10.1007/s000390050026