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Conjecture on the modulus of two-dimensional rogue waves

Determine whether the modulus of two-dimensional rogue waves in (2+1)-dimensional nonlinear wave models can be both a rational function and an exponential function, as conjectured by the authors.

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Background

The authors propose a generalization of the notion of two-dimensional rogue waves: unlike the one-dimensional case typified by the Peregrine soliton of the nonlinear Schrödinger equation, they conjecture that in (2+1)-dimensional models the modulus of rogue waves may take either rational or exponential functional forms.

Within the (2+1)-dimensional KdV equation, they construct examples consistent with this conjecture: line-soliton- and dromion-induced rogue waves with exponential decay, and lump-induced rogue waves with algebraic (rational) decay, while noting the broader claim has not been previously reported.

References

But we will generalize their modulus can be both a rational function and an exponential function. This is a bold conjecture that, as far as we know, does not seem to to have been reported in the literature.

— Rogue waves excitation on zero-background in the (2+1)-dimensional KdV equation (2405.11228 - Zhang et al., 18 May 2024) in Introduction (immediately before Eq. (1))