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<p>Generalized coupled dispersionless equations describe the dynamics of a current-fed string in an external magnetic field. In this study, we propose a novel parity-time symmetric, reverse space-time, nonlocal semi-discrete complex coupled dispersionless system which is derived from the discretization of the corresponding Lax pair. Using the Darboux transformation, we construct multi-soliton solutions represented as determinant ratios. Explicit expressions for one- and two-soliton solutions are obtained to analyze the system’s behavior. This analysis is further refined using a Bayesian regularization backpropagation artificial neural network, which was rigorously validated through relative <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11216_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-errors during training and testing on clean and noisy data. The validation process included detailed tabular and graphical representations, confirming the reliability of the analytical results. Our study also explores both symmetry-preserving and symmetry-breaking solutions within the proposed parity-time symmetric, reverse space-time, nonlocal semi-discrete complex coupled dispersionless system.</p>

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\(\mathscr{P}\mathscr{T}\)-Symmetry in nonlocal spatial discrete complex coupled dispersionless system: analytical and computational insights

  • 期刊论文

  • 发表日期：

    2025年4月28日

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  • H. W. A. Riaz,
  • Aamir Farooq,
  • J. Lin

摘要

Generalized coupled dispersionless equations describe the dynamics of a current-fed string in an external magnetic field. In this study, we propose a novel parity-time symmetric, reverse space-time, nonlocal semi-discrete complex coupled dispersionless system which is derived from the discretization of the corresponding Lax pair. Using the Darboux transformation, we construct multi-soliton solutions represented as determinant ratios. Explicit expressions for one- and two-soliton solutions are obtained to analyze the system’s behavior. This analysis is further refined using a Bayesian regularization backpropagation artificial neural network, which was rigorously validated through relative \(L_2\) L 2 -errors during training and testing on clean and noisy data. The validation process included detailed tabular and graphical representations, confirming the reliability of the analytical results. Our study also explores both symmetry-preserving and symmetry-breaking solutions within the proposed parity-time symmetric, reverse space-time, nonlocal semi-discrete complex coupled dispersionless system.