Reply: On Role of Symmetries in Kelvin Wave Turbulence

In the Ref. (Lebedev and L’vov in J. Low Temp. Phys. 161, 2010 , doi: 10.1007/s10909-010-0215-2 ), this issue, two of us (VVL and VSL) considered symmetry restriction on the interaction coefficients of Kelvin waves and demonstrated that linear in small wave vector asymptotic, obtained analytically, is not forbidden, as Kosik and Svistunov (KS) expect by naive reasoning. Here we discuss this problem in additional details and show that theoretical objections by KS, presented in Ref. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010 , doi: 10.1007/s10909-010-0242-z ), this issue, are irrelevant and their recent numerical simulation, presented in Ref. (Kozik and Svistunov in arXiv:1007.4927v1 , 2010 ) is hardly convincing. There is neither proof of locality nor any refutation of the possibility of linear asymptotic of interaction vertices in the KS texts, Refs. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010 , doi: 10.1007/s10909-010-0242-z ; arXiv:1006.0506v1 , 2010 ). Therefore we can state again that we have no reason to doubt in this asymptote, that results in the L’vov–Nazarenko energy spectrum of Kelvin waves.