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Register a new userOn the Zakharov-Lvov stochastic model for wave turbulence
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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.
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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.
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.
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
We obtain certain Mellin-Barnes integrals which present wave functions for $GL(n,mathbb{R})$ hyperbolic Sutherland model with arbitrary positive coupling constant.
We continue our exercises with the universal $R$-matrix based on the Khoroshkin and Tolstoy formula. Here we present our results for the case of the twisted affine Kac--Moody Lie algebra of type $A^{(2)}_2$. Our interest in this case is inspired by the fact that the Tzitzeica equation is associated with $A^{(2)}_2$ in a similar way as the sine-Gordon equation is related to $A^{(1)}_1$. The fundamental spin-chain Hamiltonian is constructed systematically as the logarithmic derivative of the transfer matrix. $L$-operators of two types are obtained by using q-deformed oscillators.
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