On two nonintegrable cases of the generalized Hénon-Heiles system

Vernov, S. Yu.; Timoshkova, E. I.

doi:10.1134/1.2131124

Cited by 6 publications

(3 citation statements)

References 19 publications

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“…For some values of parameters the obtained solutions coincide with the known exact periodic solutions. The Painlevé test does not show any obstacle to the existence of three-parameter single-valued solutions, so, the probability to find exact, for example elliptic, three-parameter solutions, that generalize the solutions found in [9], is high. The author is grateful to R. Conte …”

Section: Resultsmentioning

confidence: 99%

“…Moreover the function y, solution of system (2), satisfies the following fourth-order equation, which does not include µ: ), and also C = −2, in which these two Cases coincide. It has been shown in [8] (for µ = 0) and [9] (for an arbitrary value of µ) that single-valued three-parameter special solutions can exist only in two nonintegrable cases: C = −16/5 and C = −4/3 (λ is arbitrary).…”

Section: The Hénon-heiles Hamiltonianmentioning

confidence: 99%

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Construction of Solutions for Nonintegrable Systems with the Help of the Painlevé Test

Vernov

2004

Computational Science - ICCS 2004

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Abstract. The generalized Hénon-Heiles system with an additional nonpolynomial term has been considered. In two nonintegrable cases with the help of the Painlevé test new special solutions have been found as converging Laurent series, depending on three parameters. For some values of these parameters the obtained Laurent series coincide with the Laurent series of the known elliptic solutions. The calculations have been made with use of computer algebra system REDUCE. The obtained local solutions can assist to find the elliptic three parameters solutions. The corresponding algorithm has been realized in REDUCE and Maple. The Painlevé TestWhen we study some mechanical problem time is assumed to be real, whereas the integrability of motion equations is connected with the behavior of their solutions as functions of complex time. Solutions of a system of ODE's are regarded as analytic functions, maybe with isolated singular points. A singular point of a solution is said critical (as opposed to noncritical) if the solution is multivalued (single-valued) in its neighborhood and movable if its location depends on initial conditions. The general solution of an ODE of order N is the set of all solutions mentioned in the existence theorem of Cauchy, i.e. determined by the initial values. It depends on N arbitrary independent constants. A special solution is any solution obtained from the general solution by giving values to the arbitrary constants. A singular solution is any solution which is not special, i.e. which does not belong to the general solution. A system of ODE's has the Painlevé property if its general solution has no movable critical singularity point [1].The Painlevé test is any algorithm, which checks some necessary conditions for a differential equation to have the Painlevé property. The original algorithm, developed by P. Painlevé and used by him to find all the second order ODE's with Painlevé property, is known as the α-method. The method of S.V. Kovalevskaya [2] is not as general as the α-method, but much more simple. The remarkable property of this test is that it can be checked in a finite number of steps.

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Section: Resultsmentioning

confidence: 99%

Section: The Hénon-heiles Hamiltonianmentioning

confidence: 99%

Construction of Solutions for Nonintegrable Systems with the Help of the Painlevé Test

Vernov

2004

Computational Science - ICCS 2004

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“…At present methods for construction of special solutions of nonintegrable systems in terms of elementary (more precisely, degenerated elliptic) and elliptic functions are actively developed [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] (see also [21] and references therein). Some of these methods are intended for the search for elliptic solutions only [11,15]; others allow us to find either solutions in terms of elementary functions only [2][3][4]20] or both types of solutions [1, 5-10, 12-14, 16-19].…”

Section: Introductionmentioning

confidence: 99%

Elliptic solutions of the quintic complex one-dimensional Ginzburg–Landau equation

Vernov¹

2007

J. Phys. A: Math. Theor.

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The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows us to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5. Using the investigation of the CGLE5 as an example, we demonstrate that to find elliptic solutions the analysis of a system of differential equations is more preferable than the analysis of the equivalent single differential equation.

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Interdependence Between the Laurent-Series and Elliptic Solutions of Nonintegrable Systems

Vernov

2005

Computer Algebra in Scientific Computing

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On two nonintegrable cases of the generalized Hénon-Heiles system

Cited by 6 publications

References 19 publications

Construction of Solutions for Nonintegrable Systems with the Help of the Painlevé Test

Construction of Solutions for Nonintegrable Systems with the Help of the Painlevé Test

Elliptic solutions of the quintic complex one-dimensional Ginzburg–Landau equation

Interdependence Between the Laurent-Series and Elliptic Solutions of Nonintegrable Systems

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