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Granular chain wave dynamics under Hertz nonlinearity via a hybrid Hirota-based bilinear neural network

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  • Ali, Asad
  • Hassan, Azrar Ul
  • Rizvi, Syed Tahir Raza

Abstract

This paper explores the nonlinear wave dynamics of a granular chain of elastic spheres (GCES) with a hybrid bilinear neural network (BNNA) with a double-hidden layer neural network (DHLNN) architecture. The long-wave structure of the GCES wave model is obtained, and the Hertz model is used to include the nonlinearity inherent in the granular medium. The BNNA framework is developed through the combination of the Hirota bilinear transformation (HBT) and the Hilbert transformation to build the analytical wave solutions. With BNNA, a number of localized and interaction patterns are obtained, such as lump kink soliton (LKS), lump stripe solutions (LSSs), lump fusion with triangular periodic (IBLTP) waves, interaction of lump, periodic, and 1-kink soliton (IBLP1KS) waves, and breather wave (BW)-type structures. Surface and contour plots are used to investigate the spatiotemporal dynamics of these patterns, and they display rich behaviors of localization, interaction, and transfer. The findings affirm that BNNA is an effective analytical-computational method for modeling nonlinear wave dynamics in GCES and has potential in similar physical processes in optics and plasma.

Suggested Citation

  • Ali, Asad & Hassan, Azrar Ul & Rizvi, Syed Tahir Raza, 2026. "Granular chain wave dynamics under Hertz nonlinearity via a hybrid Hirota-based bilinear neural network," Chaos, Solitons & Fractals, Elsevier, vol. 208(P3).
    • Handle: RePEc:eee:chsofr:v:208:y:2026:i:p3:s0960077926004479
    DOI: 10.1016/j.chaos.2026.118306
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