| Abstract: |
| We study the nonlinear Fourier spectrum of four shallow-water wave models --KdV, Serre-Green-Naghdi (SGN), Madsen-Sorensen, and Whitham-- using the periodic inverse scattering transform (pIST) of KdV as a spectral diagnostic, justified by a piecewise KdV-coupling approximation argument. Applying this framework to numerical simulations and laboratory data of multisoliton fission from Trillo et al. (Phys. Rev. Lett., 2016), we find that SGN best reproduces the KdV-theoretical soliton-spectrum amplitudes within 3-7% RMSE, yet all models exhibit substantially larger and increasingly variable errors against laboratory data, with up to 30-60% RMSE at the highest Ursell (Ur) numbers (84--26,000). This growing model-lab discrepancy is accompanied by a spatial drift of pIST soliton amplitudes along the channel, which would remain constant under perfect KdV integrability, providing direct spectral evidence of integrability breakdown driven by dissipation. We propose a physical conjecture for the dominant dissipation mechanism, supported by fitted power-law decay of pIST soliton amplitudes along the channel. These findings establish a spectral benchmark for shallow-water models and demonstrate that dissipation closures accounting for subtle energy losses are essential to capture multisoliton fission dynamics at high Ur. |
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