**Физика Низких Температур: ***Том ***36**, Выпуск **8-9** (Август 2010), c. 986-993
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### Weak turbulence of Kelvin waves in superfluid He

**Victor S. L’vov**^{1,3} and Sergey Nazarenko^{2}

^{1}Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel

^{2}Mathematics Institute, Warwick University, Coventry, CV4 7AL, UK

^{3}Theoretical Department, Institute for Magnetism, National Academy of Sciences of Ukraine, Kiev, Ukraine

E-mail: victor.lvov@weizmann.ac.il

*Received January 21, 2010*

**Аннотация**

Physics of small-scale quantum turbulence in superfluids is essentially based on the knowledge of the energy
spectrum of Kelvin waves, *E*_{k}. In our paper, we derive a new type of kinetic equation for Kelvin waves on quantized vortex filaments with random large-scale curvature which describes a step-by-step energy cascade over scales caused by five-wave interactions. This approach replaces the previously used six-wave theory, which was recently shown to be inconsistent due to nonlocality. Solving the four-wave kinetic equation, we found a new *local* spectrum with a universal (curvature-independent) exponent,
*E*_{k} *k* µ ^{−5/3}, which must replace the nonlocal spectrum of the six-wave theory, *E*_{k} *k* µ ^{−7/5}, in future theory, e.g., in finding the quantum turbulence
decay rate, found by Kosik and Svistunov under wrong assumption of the locality of energy transfer in the sixwave interactions.

PACS: 67.25.dk Vortices and turbulence;

PACS: 47.37.+q Hydrodynamic aspects of superfluidity; quantum fluids;

PACS: 45.10.Hj Perturbation and fractional calculus methods;

PACS: 47.10.Df Hamiltonian formulations.

**Ключевые слова: **turbulence, superfluid, Kelvin wave.