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Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 548–558 (Mi mais521)  

This article is cited in 2 scientific papers (total in 2 papers)

Two wave interactions in a Fermi–Pasta–Ulam model

S. D. Glyzinab, S. A. Kashchenkoac, A. O. Tolbeya

a P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia
b Scientific Center in Chernogolovka RAS, 9 Lesnaya str., Chernogolovka, Moscow region, 142432, Russia
c National Research Nuclear University MEPhI, 31 Kashirskoye shosse, Moscow 115409, Russia

Abstract: The work is devoted to the dynamic properties of the solutions of boundary value problems associated with the classical system of Fermi–Pasta–Ulam (FPU). We study this problem in infinite-dimensional case, when a countable number of roots of characteristic equations tend to an imaginary axis. Under these conditions, we built a special non-linear partial differential equation, which plays the role of a quasinormal form, i.e, it defines the dynamics of the original boundary value problem with the initial conditions in a sufficiently small neighborhood of the equilibrium state. The modified Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) one are quasi-normal forms depending on the parameter values. Under some additional assumptions, we apply the procedure of renormalization to the obtained boundary value problems. This procedure leads to an infinite-dimensional system of ordinary differential equations. We describe a method of folding this system in the special boundary value problem, which is an analogue of the normal form. The main result is that the analytical methods of nonlinear dynamics explored the interaction of waves moving in different directions, in the problem of the FPU. It was shown that waves influence on each other is asymptotically small and does not change the shape of waves, contributing only a shift in their speed, which does not change over time.

Keywords: Fermi–Pasta–Ulam model, generalized KdV equation, quasinormal form, boundary value problem.

Funding Agency Grant Number
Russian Science Foundation 14-21-00158
This work was supported by the Russian Science Foundation (project nos. №14-21-00158).


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UDC: 517.9
Received: 15.06.2016

Citation: S. D. Glyzin, S. A. Kashchenko, A. O. Tolbey, “Two wave interactions in a Fermi–Pasta–Ulam model”, Model. Anal. Inform. Sist., 23:5 (2016), 548–558

Citation in format AMSBIB
\by S.~D.~Glyzin, S.~A.~Kashchenko, A.~O.~Tolbey
\paper Two wave interactions in a Fermi--Pasta--Ulam model
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 5
\pages 548--558

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    This publication is cited in the following articles:
    1. S. A. Kashchenko, “Families of Normalizes Equations in the Problem of Dislocations in a Solid Body”, Dokl. Math., 96:2 (2017), 517–521  crossref  mathscinet  zmath  isi  scopus
    2. S. A. Kashchenko, “Regular and irregular solutions in the problem of dislocations in solids”, Theoret. and Math. Phys., 195:3 (2018), 807–824  mathnet  crossref  crossref  adsnasa  isi  elib
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