Fractional-order bilinear feature network for exact solutions of the (1+1)-dimensional Caudrey–Dodd–Gibbon equation
Keywords:
Caudrey–Dodd–Gibbon equation; Fractional-Order Bilinear Feature Network (FOBFN); conformable fractional derivative; Hirota bilinear method; exact soliton solutions; symbolic computation; nonlinear wave dynamicsAbstract
The fifth-order dispersive nonlinear wave model for plasma magnetosonic waves and soliton propagation in optical fibers is the (1+1)-dimensional Caudrey-Dodd-Gibbon (CDG) equation. The equation has many solution structures, but recovering them in exact closed form is an analytical difficulty. The bilinear neural network methods used are based on regular polynomial feature augmentation, which limits their ability to handle memory effects and anomalous dispersion when the CDG equation is extended to fractional order. Our Fractional-Order Bilinear Feature Network fills this gap. The concept is to replace ordinary monomial features x2, t2, and xt with conformable fractional-order features xα, tα, and (xt)α/2 with order α ∈ (0, This expands the trial function’s representation while maintaining Hirota bilinear machinery’s symbolic accuracy. Maple imposes algebraic zero-residual constraints on the fractional system and finds four closed-form precise solutions for single- and double-hidden-layer network architectures. Comparative simulations in integer and fractional regimes demonstrate the impact of α on solitons’ amplitude, propagation speed, and energy localization. Three-dimensional surface plots, contour diagrams, density maps, and multi-time curve graphs show dynamics. On the domain x,t ∈ (−30,30), numerical checks reveal 2-4% enhancements in peak amplitude and wave-front sharpness fidelity over the conventional integer-order baseline, with no analytical error. The framework gives an easy technique to create accurate solutions of fractional CDG variations arising in plasma physics, quantum field theory and nonlinear optics.
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