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Volume 38 (2025)
Volume 38, Issue 1
Pg 1 - 157 (June 2025)
Volume 37 (2024)
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Pg 107 - 263 (December 2024)
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Pg 1 - 105 (June 2024)
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Pg 129 - 202 (December 2023)
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Pg 1 - 127 (June 2023)
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Pg 1 - 44 (December 2022)
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Volume 34,
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Volume 33, Issue 2
Pg 81 - 130 (December 2021)
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Pg 1 - 80 (June 2021)
Volume 32 (2020)
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Pg 67 - 138 (December 2020)
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Pg 1 - 65 (June 2020)
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Pg 63 - 106 (December 2019)
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Pg 1 - 61 (June 2019)
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Pg 135 - 209 (December 2018)
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Pg 77 - 133 (September 2018)
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Pg 37 - 75 (June 2018)
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Pg 1 - 35 (March 2018)
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Pg 137 - 168 (December 2017)
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Far East Journal of Dynamical Systems
Far East Journal of Dynamical Systems
Volume 9, Issue 2, Pages 223 - 254 (June 2007)
A PROOF THAT CERTAIN CLASSES OF INVARIANT PAINLEVÉ SOLUTIONS OF NLPDEs ARE TRAVELING WAVES AND APPLICATIONS TO FORCED INTEGRABLE AND NONINTEGRABLE SYSTEMS
S. Roy Choudhury (USA)
Abstract:
We consider a technique for deriving exact analytic coherent structure (pulse/front/domain wall) solutions of general NLPDEs via the use of truncated invariant Painlevé expansions, and prove that these solutions satisfy the corresponding traveling wave reduced ODE (as conjectured by Powell et al. [57]). Thus, they not only provide ‘partial integrability’ in Painlevé’s original sense but also a parameterization of the homoclinic or heteroclinic structures of the traveling wave reduced ODEs. Coupling this to Melnikov theory, we then consider the breakdown to chaos of such analytic coherent structure solutions of various long-wave and reaction-diffusion equations under forcing. We also demonstrate that similar treatments are possible for integrable systems (where the soliton/kink solutions represent the homoclinic/ heteroclinic structures of the reduced ODEs) using the well-studied forced sine-Gordon equation as the main example. A method of treating the dynamics of the system prior to the onset of chaos by the use of intrinsic harmonic balance, multiscale or direct soliton perturbation theory is briefly discussed. It is conceivable that resummation of such perturbation series via the use of Pade approximants or other techniques may enable one to analytically follow the homoclinic or heteroclinic tangling beyond the first transversal intersection of the stable and unstable manifolds and into the chaotic regime.
Keywords and phrases:
analytic solutions, Painlevé analysis, homoclinic and heteroclinic orbits, Melnikov analysis.
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P-ISSN: 0972-1118
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