An integrable hierarchy with a perturbed Hénon–Heiles system

Hone, Andrew N. W.; Novikov, V. S.; Verhoeven, Caroline

doi:10.1088/0266-5611/22/6/006

Cited by 18 publications

(20 citation statements)

References 48 publications

(92 reference statements)

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“…In a more general direction Berkovich applied his method of factorisation [10,11] to problems based upon Ermakov systems. Further applications are found in Cosmology [39-41, 75, 94, 95, 97, 99, 100], partial differential equations of Mathematical Physics such as the Korteweg -de Vries and Camassa -Holm equations [19,20,22,[42][43][44][45][46], Elasticity [77-79, 94, 99], Quantum Mechanics [48] and nonlinear systems in general [15,18,25,47,50,51,63,64].…”

Section: Ermakov and His Invariantmentioning

confidence: 99%

The Ermakov equation: A commentary

Leach

Andriopoulos

2008

APPL ANAL DISCRETE M

112

127

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We present a short history of the Ermakov Equation with an emphasis on its discovery by the West and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008), 123-145) for an English translation of Ermakov's original paper. THE LEWIS INVARIANT AND PINNEY'S SOLUTIONIn 1950 the late Edmund Pinney [2] presented in a very succinct paper [85] the solution of the equationin which the overdot represents differentiation with respect to the independent variable, t, which in many applications is the time (See also [7]). The solution which he gave is(2) x(t) = Au 2 + 2Buv + Cv 2 1/2 , where u(t) and v(t) are any two linearly independent solutions of the equation0 2000 Mathematics Subject Classification. 34A05, 01A75, 70F05, 22E70. The Ermakov Equation: history and impact 147 and the constants, A, B and C, are related according to B 2 − AC = 1/W 2 with W being the constant Wronskian of the two linearly independent solutions.In 1966 the late Ralph Lewis, while he was on sabbatical at the University of Heidelberg, commenced the calculation of an invariant for the Hamiltonian corresponding to (3), videlicetusing Kruskal's asymptotic method [54]. An adiabatic invariant for (4) had been proposed by Lorentz at the Solvay Congress of 1911 [98], but this was not satisfactory for the application of interest to Lewis which was the motion of a charged particle in an electromagnetic field expected to be rapidly varying in the context of plasma confinement. Kruskal's method involves an asymptotic expansion in terms of a parameter, ε, and for the first term in the expansion Lewis obtainedwhere ρ(t) is a solution of (1) and so is given by (2). To the surprise of Lewis the second and third terms in the asymptotic expansion were zero. He then essayed 1 a direct calculation of the first derivative of I 0 and found it to be zero! His results are found in [65][66][67].Although this result could scarcely be considered to be of use in classical problems, the reduction of the solution of the corresponding time-dependent Schrödinger equation [68] to the solution of (3) could be regarded as a distinct advantage since any numerical computation was then deferred until almost the last line. A clear example of this is found in the calculation of Berry's Phase for the time-dependent linear oscillator [55]. The utility of the result was enhanced when it was found that (1) occurred as an auxiliary equation in the calculation of invariants for nonquadratic Hamiltonian systems [16,17,37,38,26,[69][70][71][72][73][74]. ERMAKOV AND HIS INVARIANTIn Ermakov's paper the roles of (1), (3) and (5) are interchanged by comparison with the work of Lewis 2 . Ermakov introduces (1) as an equation auxiliary to (3), multiplies by an integr...

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Section: Ermakov and His Invariantmentioning

confidence: 99%

The Ermakov equation: A commentary

Leach

Andriopoulos

2008

APPL ANAL DISCRETE M

112

127

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“…System (99) has been considered independently by authors of [59,60] and [61], where the corresponding Lax representation has been obtained. The Lax representations for equation (98) is still not known.…”

Section: Propositionmentioning

confidence: 99%

Symbolic representation and classification of integrable systems

Михайлов

Новиков

Wang

2008

Algebraic Theory of Differential Equations

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“…In this case, a scheme similar to that described above for constructing particular solutions can be found in [7] (also see [4]). Additional possibilities concerning the classification of polynomial solutions G(λ) appear if we note that map (4.1) is factorable.…”

Section: The Map G → Umentioning

confidence: 99%

Model equation of the theory of solitons

Адлер

Shabat

2007

Theor Math Phys

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We consider the hierarchy of integrable (1+2)-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain n arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable (1+1)-dimensional equations.

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An integrable hierarchy with a perturbed Hénon–Heiles system

Cited by 18 publications

References 48 publications

The Ermakov equation: A commentary

The Ermakov equation: A commentary

Symbolic representation and classification of integrable systems

Model equation of the theory of solitons

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