PapersNo Access

Exact Solutions in the Invariant Manifolds of the Generalized Integrable Hénon–Heiles System and Exact Traveling Wave Solutions of Klein–Gordon–Schrödinger Equations

Jibin Li

Jibin Li

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, P. R. China

E-mail Address: lijb@zjnu.cn

Search for more papers by this author

https://doi.org/10.1142/S0218127417500122Cited by:1 (Source: Crossref)

Abstract

In this paper, we consider the exact explicit solutions for the famous generalized Hénon–Heiles (H–H) system. Corresponding to the three integrable cases, on the basis of the investigation of the dynamical behavior and level curves of the planar dynamical systems, we find all possible explicit exact parametric representations of solutions in the invariant manifolds of equilibrium points in the four-dimensional phase space. These solutions contain quasi-periodic solutions, homoclinic solutions, periodic solutions as well as blow-up solutions. Therefore, we answer the question: what are the flows in the center manifolds and homoclinic manifolds of the generalized Hénon–Heiles (H–H) system. As an application of the above results, we consider the traveling wave solutions for the coupled $(n + 1)$ $(n + 1)$ -dimensional Klein–Gordon–Schrödinger Equations with quadratic power nonlinearity.

This research was partially supported by the National Natural Science Foundation of China (11471289, 11571318).

Keywords:

References

Arai, A. [1991] “ Exactly solvable supersymmetric quantum mechanics,” J. Math. Anal. Appl. 158, 63–79. Crossref, Web of Science, Google Scholar
Ballesteros, Á. & Blasco, A. [2010] “ Integrable Hénon–Heiles Hamiltonians: A Poisson algebra approach,” Ann. Phys. 325, 2787–2799. Crossref, Web of Science, Google Scholar
Bountis, T., Segur, H. & Vivaldi, F. [1982] “ Integrable Hamiltonian systems and the Painlevé property,” Phys. Rev. A 25, 1257–1264. Crossref, Web of Science, Google Scholar
Byrd, P. F. & Fridman, M. D. [1971] Handbook of Elliptic Integrals for Engineers and Scientists (Springer, Berlin). Crossref, Google Scholar
Chazy, J. [1911] “ Sur les équations difféntielles du troisiéme ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes,” Acta Math. 34, 317–385. Crossref, Google Scholar
Conte, R., Musette, R. & Verhoeven, C. [2005] “ Explicit integration of the Hénon–Heiles Hamiltonians,” J. Nonlin. Math. Phys. 12, 212–227. Crossref, Web of Science, Google Scholar
Cosgrove, C. M. [2000] “ Higher-order Painlevé equations in the polynomial class. I. Bureau symbol P2,” Stud. Appl. Math. 104, 1–65. Crossref, Web of Science, Google Scholar
Eilbeck, J. C. & Enol’skii, V. Z. [1994a] “ Elliptic solutions and blow-up in an integrable Hénon–Heiles system,” Proc. Roy. Soc. Edinburgh A 124, 1151–1164. Crossref, Web of Science, Google Scholar
Eilbeck, J. C. & Enol’skii, V. Z. [1994b] “ Elliptic Baker–Akhiezer functions and an application to an integrable dynamical system,” J. Math. Phys. 35, 1192–1201. Crossref, Web of Science, Google Scholar
Ford, J. [1973] “ Transition from analytic dynamics to statistical mechanics,” Adv. Chem. Phys. 24, 155. Crossref, Google Scholar
Gavrilov, L. [1989] “ Bifurcations of invariant manifolds in the generalized Hénon–Heiles system,” Physica D 34, 223–239. Crossref, Web of Science, Google Scholar
Grammaticos, B., Dorizzi, B. & Padjen, R. [1982] “ Painleve property and integrals of motion for the Hénon–Heiles system,” Phys. Lett. A 89, 111–113. Crossref, Web of Science, Google Scholar
Hayashi, N. & Von Wahl, W. [1987] “ On the global strong solutions of coupled Klein–Gordon–Schrödinger equations,” J. Math. Soc. Japan 39, 489–497. Crossref, Web of Science, Google Scholar
Hénon, M. & Heiles, C. [1964] “ The applicability of the third integral of motion: Some numerical experiments,” Astron. J. 69, 73–79. Crossref, Web of Science, Google Scholar
Kostov, N. A., Gerdjikov, V. S. & Mioc, V. [2010] “ Exact solutions for a class of integrable Hénon–Heiles-type systems,” J. Math. Phys. 51, 022702. Crossref, Web of Science, Google Scholar
Li, W. & Shi, S. [2011] “ Non-integrability of Hénon–Heiles system,” Celest. Mech. Dyn. Astr. 109, 1–12. Crossref, Web of Science, Google Scholar
Llibre, J. & Jiménez-Lara, L. [2011] “ Periodic orbits and non-integrability of Hénon–Heiles systems,” J. Phys. A: Math. Theor. 44, 205103-1–14. Crossref, Web of Science, Google Scholar
Noid, D. W., Koszykowski, M. L. & Marcus, R. A. [1979] “ Semiclassical calculation of bound states in multidimensional systems with Fermi resonance,” J. Chem. Phys. 71, 2864–2873. Crossref, Web of Science, Google Scholar
Ohya, M. [1996] “ Stability of stationary states for coupled Klein–Gordon–Schrödinger equations,” Nonlin. Anal. 27, 455–461. Crossref, Web of Science, Google Scholar
Ravoson, V., Gavrilov, L. & Caboz, R. [1993] “ Separability and Lax pairs for Hénon–Heiles system,” J. Math. Phys. 34, 2385–2393. Crossref, Web of Science, Google Scholar
Verhoevena, C., Musetteb, M. & Conte, R. [2002] “ Integration of a generalized Hénon–Heiles Hamiltonian,” J. Math. Phys. 43, 1906–1915. Crossref, Web of Science, Google Scholar
Wojciechowski, S. [1984] “ Separability of an integrable case of the Hénon–Heiles system,” Phys. Lett. A 100, 277–278. Crossref, Web of Science, Google Scholar