TY - JOUR

T1 - Numerical analysis of a self-similar turbulent flow in Bose–Einstein condensates

AU - Semisalov, B. V.

AU - Grebenev, V. N.

AU - Medvedev, S. B.

AU - Nazarenko, S. V.

N1 - Funding Information: Sergey Nazarenko was supported by the Chaire D’Excellence IDEX (Initiative of Excellence) awarded by Université de la Côte d’Azur, France; the European Unions Horizon 2020 research and innovation programme in the framework of Marie Sklodowska-Curie HALT project (grant agreement No 823937); and the FET Flagships PhoQuS project (grant agreement No 820392). Sergey Nazarenko and Boris Semisalov were supported by the Simons Foundation Collaboration grant Wave Turbulence (Award ID 651471). Vladimir Grebenev and Boris Semisalov were partially supported by the “chercheurs invités” awards of the Fédération Doeblin FR 2800, Université de la Côte d’Azur, France. Sergey Medvedev was partially supported by CNRS “International Visiting Researcher” award and by the Ministry of Education and Science of the Russian Federation, state assignment for fundamental research (FSUS-2020-0034). Publisher Copyright: © 2021 Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/11

Y1 - 2021/11

N2 - We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose–Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum n(ω) at the zero frequency ω. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at ω=0 and a power-law asymptotic n(ω)→ω−x at ω→∞x∈R+. Finding it amounts to solving a nonlinear eigenvalue problem, i.e. finding the value x* of the exponent x for which these two boundary conditions can be satisfied simultaneously. To solve this problem we develop a new high-precision algorithm based on Chebyshev approximations and double exponential formulas for evaluating the collision integral, as well as the iterative techniques for solving the integro-differential equation for the self-similar shape function. This procedures allow to achieve a solution with accuracy ≈4.7% which is realized for x*≈1.22.

AB - We study a self-similar solution of the kinetic equation describing weak wave turbulence in Bose–Einstein condensates. This solution presumably corresponds to an asymptotic behavior of a spectrum evolving from a broad class of initial data, and it features a non-equilibrium finite-time condensation of the wave spectrum n(ω) at the zero frequency ω. The self-similar solution is of the second kind, and it satisfies boundary conditions corresponding to a nonzero constant spectrum (with all its derivative being zero) at ω=0 and a power-law asymptotic n(ω)→ω−x at ω→∞x∈R+. Finding it amounts to solving a nonlinear eigenvalue problem, i.e. finding the value x* of the exponent x for which these two boundary conditions can be satisfied simultaneously. To solve this problem we develop a new high-precision algorithm based on Chebyshev approximations and double exponential formulas for evaluating the collision integral, as well as the iterative techniques for solving the integro-differential equation for the self-similar shape function. This procedures allow to achieve a solution with accuracy ≈4.7% which is realized for x*≈1.22.

KW - Analysis of the error

KW - Bose gas

KW - Cubature formula

KW - Nonlinear spectral problem

KW - Pseudospectral method

KW - Relaxation method

KW - Wave turbulence

UR - http://www.scopus.com/inward/record.url?scp=85107600577&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2021.105903

DO - 10.1016/j.cnsns.2021.105903

M3 - Article

AN - SCOPUS:85107600577

VL - 102

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

M1 - 105903

ER -