#### For citation:

Glyzin S. D., Kashchenko S. A., Tolbey A. O. Equations with the Fermi–Pasta–Ulam and dislocations nonlinearity. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2019, vol. 27, iss. 4, pp. 52-70. DOI: 10.18500/0869-6632-2019-27-4-52-70

# Equations with the Fermi–Pasta–Ulam and dislocations nonlinearity

Issue. The class of Fermi–Pasta–Ulam equations and equations describing dislocations are investigated. Being a bright representative of integrable equations, they are of interest both in theoretical constructions and in applied research. Investigation methods. In the present work, a model combining these two equations is considered, and local dynamic properties of solutions are investigated. An important feature of the model is the fact that the inﬁnite set of characteristic numbers of the equation linearized at zero consists of purely imaginary values. Thus, the critical case of inﬁnite dimension is realized in the problem on the stability of the zero solution. In this case a special asymptotic method for construction of the so-called normalized equations is used. Using such equations, we determine the main part of the solutions of the original equation, after that we can investigate the asymptotic behavior using perturbation theory methods. Results. All solutions are naturally divided into two classes: regular solutions that smoothly depend on a small parameter entering the equation, and irregular ones, which are a superposition of functions that oscillate rapidly on a spatial variable. For each class of solutions, areas of such changes in the parameters of the equation are distinguished in which the main parts are described by diﬀerent normalized equations. Suﬃciently wide classes of such equations are presented, which include, for example, the families of the Schro¨dinger, Korteweg–de Vries, and other equations. The problem of determining such a set of parameters of the original equation for which the nonlinearity of dislocations and the nonlinearity of the FPU are comparable «in force» is considered, i.e. none of them can be neglected in the ﬁrst approximation. Discussion. It is interesting to note that for regular and irregular solutions the areas of parameters in which nonlinearities are comparable are diﬀerent. In the second case the corresponding region is much wider. The article consists of two chapters. In the ﬁrst chapter, normalized equations for regular solutions are constructed, and in the second, for irregular ones. In turn, the ﬁrst chapter is divided into three parts, in each part diﬀerent normalized equations are constructed (depending on the values of the parameters).

- Frenkel J., Kontorova T. On the theory of plastic deformation and twinning. Acad. Sci. U.S.S.R. J. Phys., 1939, vol. 1, pp. 137–149.
- Wert Charles A. and Thomson Robb M. Physics of Solids. New York: McGraw-Hill, 1964. 436 p.
- Kudryashov N.A. Analytical properties of nonlinear dislocation equation // Applied Mathematics Letters. 2017. Vol. 69. P. 29–34.
- Kudryashov N. A. From the Fermi–Pasta–Ulam model to higher-order nonlinear evolution equations // Reports on Mathematical Physics. 2016. Vol. 77, no. 1. P. 57–67.
- Fermi E., Pasta J., Ulam S. Studies of Nonlinear Problems. I: Report LA-1940. Los Alamos Scientiﬁc Laboratory of the University of California, 1955. 21 p.
- Genta T., Giorgilli A., Paleari S., Penati T. Packets of resonant modes in the Fermi–Pasta–Ulam system // Physics Letters A. 2012. Vol. 376, no. 28. P. 2038–2044.
- Kudryashov N. A. Analytical Theory of Nonlinear Differential Equations. Izhevsk: Institute of computer investigations, 2004. 360 p.
- Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M. Method for solving the Korteweg–de Vries equation // Physical Review Letters. 1967. Vol. 19, no. 19. P. 1095–1097. ISSN 0031-9007.
- Ablowitz M.J. and Segur H. Solitons and the Inverse Scattering Transform. Philadelphia, PA.: Society for Industrial and Applied Mathematics, 1981, 425 p.
- Glyzin S.D., Kashchenko S.A., Tolbey A.O. Two wave interactions in a Fermi–Pasta–Ulam model. Modeling and Analysis of Information Systems, 2016, vol. 23, no. 5, pp. 548–558 (in Russian).
- Dodd, R.K., Eilbeck J.C., Gibbon J.D., Morris H.C. Solitons and Nonlinear Wave Equations, London et al.: Academic Press, 1982. 630 pp.
- Kashchenko S.A. Normal form for the KdV–Burgers equation. Dokl. Math., 2016, vol. 93, no. 3, pp. 331–333.
- Kaschenko S. A. Normalization in the systems with small diﬀusion // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 1996. Vol. 6, no. 6. P. 1093–1109.
- Kashchenko I.S., Kashchenko S.A. Local dynamics of the two-component singular perturbed ssystems of parabolic type // International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2015. Vol. 25, no. 11. P. 1550142.
- Glyzin S.D., Kashchenko S.A., Tolbey A.O. Two-wave interactions in the Fermi–Pasta–Ulam model // Automatic Control and Computer Sciences. 2017. Vol. 51, No. 7. Pp. 627–633.
- Kaschenko S.A. Bifurcational features in systems of nonlinear parabolic equations with weak diﬀusion // International Journal of Bifurcation and Chaos. 2005. Vol. 15, no. 11. P. 3595–3606.
- Newell A.C. Solitons in Mathematics and Physics. Philadelphia, Pa.: Society for Industrial and Applied Mathematics, 1985. 260 p.
- Zabusky N.J., Kruskal M.D. Interaction of «solitons» in a collisionless plasma and the recurrence of initial states // Phys Rev. Lett. 1965. Vol. 15. P. 240–243.
- Korteweg D.J., de Vries G. On the change of form of long waves advancing in a rectangular canal and on a new tipe of long stationary waves // Phil. Mag. 1895. Vol. 39. P. 422–443.
- Burgers J.M. A mathematical model illustrating the theory of turbulence // Adv. Appl. Mech. 1948. Vol. 1. P. 171–199.
- Rabinovich M.I., Trubetskov D.I. Introduction to the Theory of Oscillations and Waves. Moscow: Nauka, 1984; Dordrecht: Kluwer, 1989.
- Kudryashov N.A. On «new travelling wave solutions» of the KdV and the KdV–Burgers equations // Commun. Nonlinear Sci. Numer. Simul. 2009. Vol. 14(5). P. 1891–1900.
- Kudryashov N.A. Exact soliton solutions of the generalized evolution equation of wave dynamics // Journal of Applied Mathematics and Mechanics. 1988. Vol. 52, no. 3. P. 361–365.
- Kudryashov N.A. One method for ﬁnding exact solutions of nonlinear diﬀerential equations // Commun. Nonlinear Sci. Numer. Simul. 2012. Vol. 17 (6). P. 2248–2253.
- Kudryashov N.A. Painleve analysis and exact solutions of the Korteweg—de Vries equation with a source // Appl. Math. Lett. 2015. Vol. 41. P. 41–45.
- Kashchenko I.S. Multistability in nonlinear parabolic systems with low diffusion. Dokl. Math., 2010, vol. 82, no. 3, pp. 878–881.
- Kashchenko S.A. Quasinormal forms for parabolic equations with small diffusion. Soviet Math. Dokl., 1988, vol. 37, no. 2, pp. 510–513.
- Kashchenko I.S., Kashchenko S.A. Quasi-normal forms of two-component singularly perturbed systems. Dokl. Math., 2012, vol. 86, no. 3, pp. 865–870.
- Glyzin S.D., Kolesov A.Yu., Rozov N.Kh. Autowave processes in continual chains of unidirectionally coupled oscillators. Selected topics of mathematical physics and analysis. MAIK Nauka/ Interperiodica. Moscow. Proc. Steklov Inst. Math., 2014, vol. 285, pp. 81–98.
- Glyzin S.D., Kolesov A.Yu., Rozov N.Kh. Buffering effect in continuous chains of unidirectionally coupled generators. Theoret. and Math. Phys., 2014, vol. 181, no. 2, pp. 1349–1366.
- Naumkin P.I. The dissipative property of a cubic non-linear Schro¨dinger equation // Izvestiya. Mathematics. 2015. Vol. 79, no. 2. P. 346–374.
- Naumkin P.I. Solution asymptotics at large times for the non-linear Schrodinger equation // Izvestiya. Mathematics. 1997. Vol. 61, no. 4. P. 757–794.

- 2190 reads