Emergence of solitonic structures in hierarchical Korteweg-de Vries systems. (English) Zbl 1475.74073
Berezovski, Arkadi (ed.) et al., Applied wave mathematics II. Selected topics in solids, fluids, and mathematical methods and complexity. Cham: Springer. Math. Planet Earth 6, 89-124 (2019).
Summary: We explore numerically different types of solutions of a hierarchical Korteweg-de Vries equation [P. Giovine and F. Oliveri, Meccanica 30, No. 4, 341–357 (1995; Zbl 0837.73018)] that describes inter alia wave propagation in microstructured (dilatant granular) materials. This equation contains three material parameters that collectively determine the type of solutions. The simulations focus on the effect the microstructure has on the field of motion driven by harmonic, cnoidal and \(\operatorname{\text{sech}}^2\)-type initial waves. The simulations employ a Fourier transform based pseudospectral method and have been performed for a wide range of material parameters. The results are interpreted in terms of the properties of Korteweg-de Vries solitons. The analysis reveals considerable evolution and transformations of the field of motions during the propagation of signals of various kind due to the effect of the microstructure. A large part of the numerically tracked solutions have properties that match the core properties of solitons. On many occasions the emerging waves in the system propagate at a constant speed, keep their shape and interact with other similar entities elastically. The number of emerging solitons markedly depends on the ratio of dispersion parameters for micro- and macrostructure, and on the shape of the initial wave.
For the entire collection see [Zbl 1443.35005].
For the entire collection see [Zbl 1443.35005].
MSC:
| 74J35 | Solitary waves in solid mechanics |
| 74E20 | Granularity |
| 74M25 | Micromechanics of solids |
| 74S25 | Spectral and related methods applied to problems in solid mechanics |
Keywords:
microstructured dilatanant granular material; material parameter; wave propagation; pseudospectral method; Fourier transformCitations:
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