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Письма в Журнал экспериментальной и теоретической физики  / №1 2017

FERMI–PASTA–ULAM RECURRENCE AND MODULATION INSTABILITY (200,00 руб.)

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Первый авторKuznetsov
Страниц2
ID581092
Аннотация1. The phenomenon of recurrence in nonlinear systems with many degrees of freedom was first observed in the classical numerical experiments by and Ulam [1] in 1954. The idea of Fermi was to address how randomization due to the nonlinear interaction leads to the energy equipartition between large number of degrees of freedom in the mechanical chain. The chain in [1] had a quadratic nonlinearity and included 64 oscillators supplemented with long-wave initial con- ditions. Instead of the anticipated energy equipartition, numerical experiments showed that after a finite time a recurrence to the initial data was achieved accompanied by a quasi-periodic energy exchange between several initially exited modes. That recurrence phenomenon be- came known as the famous problem and has been one the nonlinear science, studied in numerous papers. Many other peculiarities of this problem have been discovered Fermi–Pasta–Ulam (FPU) of the central problems of and studied (for more details, see the original papers [2] published before the era of integrability for nonlinear
Kuznetsov, E.A. FERMI–PASTA–ULAM RECURRENCE AND MODULATION INSTABILITY / E.A. Kuznetsov // Письма в Журнал экспериментальной и теоретической физики .— 2017 .— №1 .— С. 109-110 .— URL: https://rucont.ru/efd/581092 (дата обращения: 28.04.2025)

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The phenomenon of recurrence in nonlinear systems with many degrees of freedom was first observed in the classical numerical experiments by Fermi, Pasta, and Ulam [1] in 1954. <...> The idea of Fermi was to address how randomization due to the nonlinear interaction leads to the energy equipartition between large number of degrees of freedom in the mechanical chain. <...> The chain in [1] had a quadratic nonlinearity and included 64 oscillators supplemented with long-wave initial conditions. <...> Instead of the anticipated energy equipartition, numerical experiments showed that after a finite time a recurrence to the initial data was achieved accompanied by a quasi-periodic energy exchange between several initially exited modes. <...> That recurrence phenomenon became known as the famous Fermi–Pasta–Ulam (FPU) problem and has been one of the central problems of the nonlinear science, studied in numerous papers. <...> Many other peculiarities of this problem have been discovered and studied (for more details, see the original papers [2] published before the era of integrability for nonlinear systems). <...> Since the discovery of the Inverse Scattering Transform, which was first applied to the KDV equation [3], and later to the nonlinear Schrodinger equation (NLSE) [4], many aspects of the FPU recurrence became more clear. <...> The FPU recurrence was also intensively studied experimentally (the first experimental demonstration in optical fibers of the FPU recurrence was presented in [6]). <...> The main goal of this paper is to explain how the recurrence phenomenon appears within the onedimensional NLSE iψt +ψxx +2|ψ|2ψ = 0. 1)e-mail: kuznetso@itp.ac.ru 108 (1) This equation with a reasonable accuracy describes propagation of optical solitons in fibers. <...> At the present time, there are known many exact solutions of the NLSE which describe propagation of solitons/breathers on the condensate background. <...> Such solution was constructed for a first time in [7] and later inmany other papers (see [8] and references therein). <...> All these solutions show that after a while the condensate recovers its amplitude but has a different (but constant) phase. <...> This is the analogue of the FPU recurrence for the NLSE (see recent numerical confirmations [9] for arbitrary initial conditions). <...> In this paper we give a qualitative explanation of the FPU analog for cnoidal waves. <...> To find the cnoidal wave to Eq. (1) one <...>