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Register a new userFormal expansions in stochastic model for wave turbulence 1: kinetic limit
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We consider the damped/driver (modified) cubic NLS equation on a large torus with a properly scaled forcing and dissipation, and decompose its solutions to formal series in the amplitude. We study the second order truncation of this series and prove that when the amplitude goes to zero and the torus size goes to infinity the energy spectrum of the truncated solutions becomes close to a solution of the damped/driven wave kinetic equation. Next we discuss higher order truncations of the series.
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We continue the study of small amplitude solutions of the damped/driven cubic NLS equation, written as formal series in the amplitude, initiated in our previous work [Formal expansions in stochastic model for wave turbulence 1: kinetic limit, arXiv:1907.04531]. We are interested in behaviour of the formal series under the wave turbulence limit the amplitude goes to zero, while the space-period goes to infinity
In this note we present the main results of the papers cite{DK, DK2}, dedicated to rigorous study of the limiting properties of the stochastic model for wave turbulence due to Zakharov-Lvov. Proofs of the assertions, stated below without reference, may be found in those works.
We continue our investigation of kinetic models of a one-dimensional gas in contact with homogeneous thermal reservoirs at different temperatures. Nonlinear collisional interactions between particles are modeled by a so-called BGK dynamics which conserves local energy and particle density. Weighting the nonlinear BGK term with a parameter $alphain [0,1]$, and the linearinteraction with the reservoirs by $(1-alpha)$, we prove that for all $alpha$ close enough to zero, the explicit spatially uniform non-equilibrium stable state (NESS) is emph{unique}, and there are no spatially non-uniform NESS with a spatial density $rho$ belonging to $L^p$ for any $p>1$. We also show that for all $alphain [0,1]$, the spatially uniform NESS is dynamically stable, with small perturbation converging to zero exponentially fast.
In this paper we study quantum stochastic differential equations (QSDEs) that are driven by strongly squeezed vacuum noise. We show that for strong squeezing such a QSDE can be approximated (via a limit in the strong sense) by a QSDE that is driven by a single commuting noise process. We find that the approximation has an additional Hamiltonian term.
We show that wave breaking occurs with positive probability for the Stochastic Camassa-Holm (SCH) equation. This means that temporal stochasticity in the diffeomorphic flow map for SCH does not prevent the wave breaking process which leads to the formation of peakon solutions. We conjecture that the time-asymptotic solutions of SCH will consist of emergent wave trains of peakons moving along stochastic space-time paths.
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