[Paper Review] On two nonintegrable cases of the generalized Henon-Heiles system with an additional nonpolynomial term

E. I. Timoshkova, S. Yu. Vernov|arXiv (Cornell University)|Feb 18, 2004

Nonlinear Waves and Solitons16 references4 citations

TL;DR

This paper presents two-parameter elliptic function solutions for two nonintegrable cases of the generalized Hénon-Heiles system with an additional nonpolynomial term ($\mu/(2x^2)$). Using singularity analysis and Laurent series expansions, the authors derive exact solutions that generalize known one-parameter elliptic solutions. The key contribution is the construction of new single-valued, two-parameter solutions in nonintegrable settings, suggesting the potential existence of three-parameter solutions in these cases.

ABSTRACT

The generalized Henon-Heiles system with an additional nonpolynomial term is considered. In two nonintegrable cases new two-parameter solutions have been obtained in terms of elliptic functions. These solutions generalize the known one-parameter solutions. The singularity analysis shows that it is possible that three-parameter single-valued solutions exist in these two nonintegrable cases. The knowledge of the Laurent series solutions simplifies search of the elliptic solutions and allows to automatize it.

Motivation & Objective

• To find single-valued special solutions in analytic form for nonintegrable cases of the generalized Hénon-Heiles system with an additional nonpolynomial term.
• To extend known one-parameter elliptic solutions to two-parameter solutions in nonintegrable settings.
• To investigate the possibility of three-parameter single-valued solutions using singularity analysis.
• To simplify the search for elliptic solutions through systematic use of Laurent series expansions.

Proposed method

• The generalized Hénon-Heiles Hamiltonian is extended with a nonpolynomial term $\mu/(2x^2)$, leading to a system of second-order nonlinear ODEs.
• The system is analyzed via Painlevé singularity analysis to identify integrable and nonintegrable cases.
• A generalized ansatz $y_t^2 = \tilde{\mathcal{A}}y^3 + \tilde{\mathcal{B}}y^{5/2} + \tilde{\mathcal{C}}y^2 + \tilde{\mathcal{D}}y^{3/2} + \tilde{\mathcal{E}}y + \tilde{\mathcal{G}}$ is used to seek elliptic solutions.
• The substitution $y = \varrho^2$ transforms the generalized ansatz into a quartic form in $\varrho_t^2$, enabling the use of Weierstrass elliptic functions.
• Laurent series solutions are constructed to guide and validate the ansatz, with parameters constrained by consistency conditions.
• The energy $H$ is expressed as a rational function of $\lambda_1$, $\lambda_2$, $P_0$, and auxiliary variables $S_q$, $R_q$, with explicit algebraic expressions derived for two nonintegrable cases: $C = -16/5$ and $C = -4/3$.

Experimental results

Research questions

• RQ1 Can two-parameter elliptic solutions be constructed for nonintegrable cases of the generalized Hénon-Heiles system with a nonpolynomial term?
• RQ2 What conditions on the parameters $C$, $\lambda_1$, $\lambda_2$, and $\mu$ allow for such solutions to exist?
• RQ3 To what extent do Laurent series solutions support the existence of three-parameter single-valued solutions in nonintegrable cases?
• RQ4 How does the inclusion of the nonpolynomial term $\mu/(2x^2)$ affect the integrability and solution structure of the system?
• RQ5 Can the ansatz $y_t^2 = \tilde{\mathcal{A}}y^3 + \tilde{\mathcal{B}}y^{5/2} + \cdots$ yield consistent, closed-form solutions in nonintegrable settings?

Key findings

• Two-parameter elliptic solutions are successfully constructed for the nonintegrable case $C = -16/5$ with $\lambda_2 = 1$, $\lambda_1$ arbitrary, and $\mu$ arbitrary.
• For $C = -4/3$, two-parameter solutions are derived with explicit expressions for $\tilde{\mathcal{A}}$, $\tilde{\mathcal{B}}$, $\tilde{\mathcal{C}}$, $\tilde{\mathcal{D}}$, $\tilde{\mathcal{E}}$, and $H$, involving square roots of quadratic forms in $\lambda_1$, $\lambda_2$, and $P_0$.
• The energy $H$ is expressed as a rational function of $\lambda_1$, $\lambda_2$, $P_0$, and auxiliary variables $S_q$ and $R_q$, with $S_q = \pm \sqrt{35(2048\lambda_1^2 - 1280\lambda_1\lambda_2 + 387\lambda_2^2)}$ and $R_q = \pm \sqrt{7(1216\lambda_1^2 - 1824\lambda_1\lambda_2 + 783\lambda_2^2)}$.
• The Laurent series analysis suggests the potential existence of three-parameter single-valued solutions in the two nonintegrable cases, extending beyond the known two-parameter solutions.
• The method of using Laurent series to guide the ansatz significantly simplifies the search for elliptic solutions and enables automation of the solution-finding process.
• The derived solutions generalize previously known one-parameter elliptic solutions and are valid in nonintegrable settings where general analytic solutions remain unknown.