Exact waveguide backbones for sparse precursors to tropical-cyclone rapid intensification

Sparse precursors to tropical-cyclone rapid intensification are hard to interpret because a slowly evolving balanced signal can coexist with faster vortex-core adjustment. We test whether this projected slow component should be recov- ered as a data-only interpolant or as a solution-audited nonlinear waveguide backbone with a weak transported correction. From trapped-mode baroclinic quasi-geostrophic potential-vorticity dynamics on a weakly inhomogeneous beta- plane, we derive coefficient-closed generalized perturbed Korteweg–de Vries and weakly transverse generalized perturbed Kadomtsev–Petviashvili equations within one multiple-scales hierarchy. Hirota bilinearization gives exact soliton tau-function families. A normalized residual audit verifies the exponential fam- ilies to roundoff and rejects previously reported one-dimensional polynomial and rational lump-type ansätze for this reduced equation. Bounded algebraic localization appears only after weak transverse dynamics are retained. The one-soliton identities yield amplitude–speed–width diagnostics and uncertainty- based acceptance rules. In sparse Hovmöller tests, a gpKdV physics-informed neural network reduces mean field error from 0.103 to 0.059. The contribution is a coefficient-closed, solution-audited slow-waveguide prior for sparse precursor reconstruction, not a convective-core rapid-intensification model.