Nonlinear Evolution Equations and Painlevé Test

Steeb, Willi-Hans; Euler, Norbert

doi:10.1142/0723

Cited by 109 publications

(71 citation statements)

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“…It seems that constraint (15) plays a central role in nonintegrable field equations (Steeb and Euler 1987). We also find this constraint for the three-velocity model.…”

Section: Carle Man and Mckean Modelssupporting

confidence: 70%

“…The constraints (see equation 15) which one finds at the resonance are the same for all three models. This constraint also appears in various nonintegrable relativistic field equations (Steeb and Euler 1988). Furthermore, the models studied do not admit Lie Backlund vector fields.…”

Section: Discussionmentioning

confidence: 93%

“…System ( (Steeb and Euler 1987). This hierarchy of Lie Backlund vector fields indicates that the system (6), with kl = k2 = 0, must be completely integrable (Steeb and Euler 1987).…”

Section: Carle Man and Mckean Modelsmentioning

confidence: 99%

“…Before studying the one-dimensional Broadwell model let us summarise the results for the two-velocity models given above (Steeb and Euler 1987). The Carleman and McKean models can be combined to the system of partial differential equations (6) where kl' k2 and k3 are constants.…”

Section: Carle Man and Mckean Modelsmentioning

confidence: 99%

“…The Painleve test (Weiss et al 1983;Steeb and Euler 1988) for the models given by equations (2), (3) and (4) is performed and the connection with integrability is discussed. Furthermore we give solutions.…”

Section: Introductionmentioning

confidence: 99%

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Painleve Test and Discrete Boltzmann Equations

Euler¹,

Steeb²

1989

Aust. J. Phys.

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Aust. J. Phys., 1989, 42, 1-10 The Painleve test for various discrete Boltzmann equations is performed. The connection with integrability is discussed. Furthermore the Lie symmetry vector fields are derived and group-theoretical reduction of the discrete Boltzmann equations to ordinary differentiable equations is performed. Lie Backlund transformations are gained by performing the Painleve analysis for the ordinary differential equations.

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“…It seems that constraint (15) plays a central role in nonintegrable field equations (Steeb and Euler 1987). We also find this constraint for the three-velocity model.…”

Section: Carle Man and Mckean Modelssupporting

confidence: 70%

Section: Discussionmentioning

confidence: 93%

“…System ( (Steeb and Euler 1987). This hierarchy of Lie Backlund vector fields indicates that the system (6), with kl = k2 = 0, must be completely integrable (Steeb and Euler 1987).…”

Section: Carle Man and Mckean Modelsmentioning

confidence: 99%

Section: Carle Man and Mckean Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Painleve Test and Discrete Boltzmann Equations

Euler¹,

Steeb²

1989

Aust. J. Phys.

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A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie‐group method

El-Sayed

Moatimid

Moussa

et al. 2014

Math Methods in App Sciences

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The Painlevé‐Kowalevski and Poly‐Painlevé Tests for Integrability

Kruskal

Clarkson

1992

Stud Appl Math

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The characteristic feature of the so-called Painleve test for integrability of an ordinary (or partial) analytic differential equation, as usually carried out, is to determine whether all its solutions are single-valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly-Pain/eve method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.

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Nonlinear Evolution Equations and Painlevé Test

Cited by 109 publications

References 0 publications

Painleve Test and Discrete Boltzmann Equations

Painleve Test and Discrete Boltzmann Equations

A study of integrability and symmetry for the (p + 1)th Boltzmann equation via Painlevé analysis and Lie‐group method

The Painlevé‐Kowalevski and Poly‐Painlevé Tests for Integrability

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