TY - JOUR

T1 - Numerical analysis of the kinetic equation describing isotropic 4-wave interactions in non-linear physical systems

AU - Semisalov, B. V.

AU - Medvedev, S. B.

AU - Nazarenko, S. V.

AU - Fedoruk, M. P.

PY - 2024/6

Y1 - 2024/6

N2 - We develop a numerical method for solving kinetic equations (KEs) that describe out-of-equilibrium isotropic nonlinear four-wave interactions in optics, deep-water wave theory, physics of superfluids and Bose gases, and in other applications. High complexity of studying numerically the wave kinetics in these applications is related with the multi-scale nature of turbulence and with power-law behaviour of turbulent spectra in the Fourier space. When solving the Cauchy problem for KE, this leads to emergence of spectra with extremely steep gradients and to occurrence of singular points in the collision integral standing in the right-hand side. To solve these problems, we develop special fast-convergent cubature formulas, highly-accurate rational approximations of the KE solution, and a new stable method for time marching. We apply the developed methods for solving the test problems of integration arisen from applications, for studying the wave kinetics in random fiber lasers and for analysing the Bose–Einstein condensation. In these applications we used KEs obtained from the Ginzburg–Landau and from the Gross–Pitaevskii equations.

AB - We develop a numerical method for solving kinetic equations (KEs) that describe out-of-equilibrium isotropic nonlinear four-wave interactions in optics, deep-water wave theory, physics of superfluids and Bose gases, and in other applications. High complexity of studying numerically the wave kinetics in these applications is related with the multi-scale nature of turbulence and with power-law behaviour of turbulent spectra in the Fourier space. When solving the Cauchy problem for KE, this leads to emergence of spectra with extremely steep gradients and to occurrence of singular points in the collision integral standing in the right-hand side. To solve these problems, we develop special fast-convergent cubature formulas, highly-accurate rational approximations of the KE solution, and a new stable method for time marching. We apply the developed methods for solving the test problems of integration arisen from applications, for studying the wave kinetics in random fiber lasers and for analysing the Bose–Einstein condensation. In these applications we used KEs obtained from the Ginzburg–Landau and from the Gross–Pitaevskii equations.

KW - Bose gas

KW - Cauchy problem

KW - Collocation method

KW - Cubature formula

KW - Inverse cascade of particles

KW - Kinetic equation

KW - Random fiber lasers

KW - Rational approximations

KW - Relaxation method

KW - Wave turbulence

UR - https://www.mendeley.com/catalogue/0b2bd6a5-0498-3a2f-9046-4de0615d11f5/

U2 - 10.1016/j.cnsns.2024.107957

DO - 10.1016/j.cnsns.2024.107957

M3 - Article

VL - 133

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

M1 - 107957

ER -