Modelling of KdV-Soliton Through Fractional Action and Emergence of Lump Waves

• Rami Ahmad El-Nabulsi ^[1]
  1. [1] Biology Centre of the Czech Academy of Sciences
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01141-6
• Enlaces
  • Texto completo
• Resumen

  • The fractional calculus of variations is considered today as an important in applied mathematics. It consists of minimizing or maximizing functionals that depend on different types of fractional derivatives and integral operators. This mathematical subject has proved to be relevant because of its motivating implications in describing dissipative and nonconservative physical systems ranging from classical to quantum mechanics and field theories. However, in some complex dynamical systems, the analytic solutions of the consequential fractional Euler–Lagrange equation are tricky to obtain, and, besides, the generalized fractional variational problems require tricky boundary conditions and further properties in order to obtain the necessary optimality conditions of the Euler–Lagrange type for the given problem. Advanced methods in the fractional calculus of variations have been found to be very practical in studying a large number of nonlinear dynamical systems; however, describing some nonlinear wave phenomena, which are of particular relevance in various fields of science, is still not extensively elaborated in literature. In this study, a new action functional based on Kiryakova fractional operators involving the Meijer-G functions is introduced and discussed. This approach was found to be practical to study periodic dampened oscillators exhibiting traveling soliton-like solutions and a family of modified Korteweg-de Vries equations and lumps obtained within the framework of weakly nonlinear dispersive differential equations. We extended our approach to the case of time-fractal Kiryakova fractional-action integral, where a set of time-fractal Korteweg-de Vries equations exhibiting lump-like evanescent wave solutions is obtained. Our approach may be applied to other nonlinear dynamical systems.

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