Bäcklund Transformations for First and Second Painlevé Hierarchies

Sakka, A.

doi:10.3842/sigma.2009.024

Cited by 3 publications

(2 citation statements)

References 22 publications

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“…The result of Theorem 5.1 can be extended to the entire orbits of parameters using Bäcklund transformations and other special solutions recently found by Sakka [24,25]. This issue will be addressed elsewhere.…”

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confidence: 85%

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

Christov¹,

Georgiev²

2015

SIGMA

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Abstract. In this paper we study the equationwhich is one of the higher-order Painlevé equations (i.e., equations in the polynomial class having the Painlevé property). Like the classical Painlevé equations, this equation admits a Hamiltonian formulation, Bäcklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters γ/λ = 3k, γ/λ = 3k − 1, k ∈ Z, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the P II -hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.

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confidence: 85%

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

Christov¹,

Georgiev²

2015

SIGMA

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“…Such equation passes the necessary condition for absence of movable branches points, called Painlevé property, and it can reduce to two Painlevé equation by appropriate choices of the parameters. The Painlevé equations are second-order nonlinear ODEs (ordinary differential equations) which define new transcendental functions [9] and it has motivated many studies in higher order ODEs [7,8,10,11,12,13,14,15,16,17,18,19,20]. Due the connection with two Painlevé equations, Kudryashov's equation was conjectured as a possibility of defining a new transcendental function [8,21].…”

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confidence: 99%

On the mKdV–Liouville hierarchy and its self-similarity reduction

Ruy

2015

Communications in Nonlinear Science and Numerical Simulation

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Integrable mixed models have been used as a generalization of traditional integrable models. However, a map from a traditional integrable model to a mixed integrable model is not well understood yet. Here, it is studied the relation between the mKdV-Liouville hierarchy and the mKdV hierarchy by employing an extended version of the modified truncation approach. This paper shows some solutions for the mKdV-Liouville hierarchy constructed from the soliton solutions of the mKdV hierarchy. The last section deals with the possibility of define new transcendental functions from the self-similarity reduction of the mKdV-Liouville hierarchy.

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Auto-Bäcklund transformations for a differential-delay equation

Gordoa

Pickering

2013

Journal of Mathematical Physics

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Discrete Painlevé equations have, over recent years, generated much interest. One property of such equations that is considered to be particularly important is the existence of auto-Bäcklund transformations, that is, mappings between solutions of the equation in question, usually involving changes in the values of parameters appearing as coefficients. We have recently presented extensions of discrete Painlevé equations to equations involving derivatives as well as shifts in the independent variable. Here we show how auto-Bäcklund transformations can also be constructed for such differential-delay equations. We emphasise that this is the first time that an auto-Bäcklund transformation has been given for a differential-delay equation.

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Bäcklund Transformations for First and Second Painlevé Hierarchies

Cited by 3 publications

References 22 publications

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

On the mKdV–Liouville hierarchy and its self-similarity reduction

Auto-Bäcklund transformations for a differential-delay equation

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