Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

G. L. Alfimov ; M. E. Lebedev

Russian journal of nonlinear dynamics, Tome 19 (2023) no. 4, pp. 473-506.

Résumé

We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called “codes” of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a “great part” of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Détail

Keywords: Duffing-type equation, periodic coefficients, symbolic dynamics, Smale horseshoe

Bibliographie

[1] Brazhnyi, V. A. and Konotop, V. V., “Theory of Nonlinear Matter Waves in Optical Lattices”, Mod. Phys. Lett. B, 18:14 (2004), 627–651 | DOI

[2] de Sterke, C. M., Eggleton, B. J., and Sipe, J. E., “Bragg Solitons: Theory and Experiments”, Spatial Solitons, Springer Ser. Opt. Sci., 82, eds. S. Trillo, W. Torruellas, Springer, Berlin, 2001, 169–209 | DOI

[3] Kivshar, Yu. S. and Agrawal, G. P., Optical Solitons: From Fibers to Photonic Crystals, Acad. Press, New York, 2003, 540 pp.

[4] Kartashov, Ya. V., Malomed, B. A., and Torner, L., “Solitons in Nonlinear Lattices”, Rev. Mod. Phys., 83:1 (2011), 247–306 | DOI

[5] Terracini, S. and Verzini, G., “Oscillating Solutions to Second-Order ODEs with Indefinite Superlinear Nonlinearities”, Nonlinearity, 13:5 (2000), 1501–1514 | DOI | MR | Zbl

[6] Ellero, E. and Zanolin, F., “Homoclinic and Heteroclinic Solutions for a Class of Second-Order Non-Autonomous Ordinary Differential Equations: Multiplicity Results for Stepwise Potentials”, Bound. Value Probl., 2013 (2013), 167, 23 pp. | DOI | Zbl

[7] Zanini, Ch. and Zanolin, F., “An Example of Chaos for a Cubic Nonlinear Schrödinger Equation with Periodic Inhomogeneous Nonlinearity”, Adv. Nonlinear Stud., 12:3 (2012), 481–499 | DOI | MR | Zbl

[8] Zanini, Ch. and Zanolin, F., “Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential”, Complexity, 2018 (2018), 2101482, 17 pp. | DOI | Zbl

[9] Alfimov, G. L. and Avramenko, A. I., “Coding of Nonlinear States for the Gross – Pitaevskii Equation with Periodic Potential”, Phys. D, 254 (2013), 29–45 | DOI | MR | Zbl

[10] Alfimov, G. L., Kizin, P. P., and Zezyulin, D. A., “Gap Solitons for the Repulsive Gross – Pitaevskii Equation with Periodic Potential: Coding and Method for Computation”, Discrete Contin. Dyn. Syst. Ser. B, 22:4 (2017), 1207–1229 | MR | Zbl

[11] Ufimsk. Mat. Zh., 7:2 (2015), 3–18 (Russian) | DOI | MR

[12] Lebedev, M. E., Alfimov, G. L., and Malomed, B. A., “Stable Dipole Solitons and Soliton Complexes in the Nonlinear Schrödinger Equation with Periodically Modulated Nonlinearity”, Chaos, 26:7 (2016), 073110, 12 pp. | DOI | MR | Zbl

[13] Rodrigues, A. S., Kevrekidis, P. G., Porter, M. A., Frantzeskakis, D. J., Schmelcher, P., and Bishop, A. R., “Matter-Wave Solitons with a Periodic, Piecewise-Constant Scattering Length”, Phys. Rev. A, 78:1 (2008), 013611, 10 pp. | DOI

[14] Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., 42, Springer, New York, 1983, 484 pp. | DOI | Zbl

[15] Moser, J., Stable and Random Motions in Dynamical Systems, With special emphasis on celestial mechanics. Hermann Weyl Lectures, the Institute for Advanced Study (Princeton), Ann. of Math. Stud., 77, Princeton Univ. Press, Princeton, N.J., 1973, viii, 198 pp. | Zbl

[16] Walter, W., Ordinary Differential Equations, Translated from the sixth German (1996) edition by Russell Thompson, Grad. Texts in Math., 182, Springer, New York, 1998, xii, 380 pp. | DOI | Zbl

[17] Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition, ed. M. Abramowitz, I. A. Stegun, Dover, New York, 1992, xiv, 1046 pp.