Higher-Order Nonlinear Dynamical Systems and Invariant Lagrangians on a Lie Group: The Case of Nonlocal Hunter–Saxton Type Peakons
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Rami Ahmad El-Nabulsi [1] ; Waranont Anukool [1]
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[1] Chiang Mai University
Chiang Mai University
Tailandia
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- Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
- Idioma: inglés
- Enlaces
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Resumen
A G-strand is an evolutionary map g(t, s) : R×R → G into a Lie group G that follows from the Hamilton’s principle for a certain class of G-invariant Lagrangians defined on the Lie algebra g of the group G. t and s are independent variables associated to a Ginvariant Lagrangian. The G-strand equations comprises a system of integrable partial differential equations obtained from the Euler–Poincaré variational equations coupled to an auxiliary zero curvature equation. Some of these integrable partial differential equations include the Hunter–Saxton equation that arises in the study of nematic liquid crystals and the Camassa–Holm equation that arises in modeling waves in shallow water including solitons and peakons. However, nonlocal integrable systems have attracted significant attention in recent years. In this study, we use a higher-order nonlocal operator approach to study nonlocal Hunter–Saxton type peakons. Peakonsantipeakons collision on Lie group is also analyzed and discussed. It was observed that the system of "two-peakon" collisions exhibits a kind of disordered behavior which is observed in various integrable and non-integrable nonlinear evolution dynamical systems.
