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Register a new userFormal expansions in stochastic model for wave turbulence 2: method of diagram decomposition
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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
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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.
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.
In arXiv:1201.4067 and arXiv:1611.08030, Eyink and Shi and Chibbaro et al., respectively, formally derived an infinite, coupled hierarchy of equations for the spectral correlation functions of a system of weakly interacting nonlinear dispersive waves with random phases in the standard kinetic limit. Analogously to the relationship between the Boltzmann hierarchy and Boltzmann equation, this spectral hierarchy admits a special class of factorized solutions, where each factor is a solution to the wave kinetic equation (WKE). A question left open by these works and highly relevant for the mathematical derivation of the WKE is whether solutions of the spectral hierarchy are unique, in particular whether factorized initial data necessarily lead to factorized solutions. In this article, we affirmatively answer this question in the case of 4-wave interactions by showing, for the first time, that this spectral hierarchy is well-posed in an appropriate function space. Our proof draws on work of Chen and Pavlovi{c} for the Gross-Pitaevskii hierarchy in quantum many-body theory and of Germain et al. for the well-posedness of the WKE.
This paper is a synopsis of the recent book A. Boritchev, S. Kuksin, textit{One-Dimensional Turbulence and the Stochastic Burgers Equation}, AMS Publications, 2021 (to appear). The book is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.
Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the equivalence generated by its equivalence group. This gives an exhaustive description of its equivalence groupoid. After extending the algebraic method of group classification to non-normalized classes of differential equations, we solve the complete group classification problem for the class under study up to both usual and general point equivalences. The solution includes the complete preliminary group classification of the class and the construction of singular Lie-symmetry extensions, which are not related to subalgebras of the equivalence algebra. The complete preliminary group classification is based on classifying appropriate subalgebras of the entire infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel invariance algebra. The results obtained can be used to construct exact solutions of nonlinear wave and elliptic equations.
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