A plethora of fully localised solitary waves for the full-dispersion Kadomtsev-Petviashvili equation

The KP-I equation arises as a weakly nonlinear model equation for gravity-capillary waves with Bond number $β>1/3$, also called strong surface tension. This equation has recently been shown to have a family of nondegenerate, symmetric fully localised' or lump’ solitary waves which decay to zero in all spatial directions. The full-dispersion KP-I equation is obtained by retaining the exact dispersion relation in the modelling from the water-wave problem. In this paper we show that the FDKP-I equation also has a family of symmetric fullly localised solitary waves which are obtained by casting it as a perturbation of the KP-I equation and applying a suitable variant of the implicit-function theorem.

💡 Research Summary

The paper investigates the full‑dispersion Kadomtsev‑Petviashvili I (FDKP‑I) equation, which incorporates the exact linear dispersion relation of three‑dimensional gravity‑capillary water waves with strong surface tension (Bond number β > 1/3). The classical Kadomtsev‑Petviashvili I (KP‑I) equation, obtained as a weakly‑nonlinear approximation, is known to possess an infinite family of symmetric, fully localized solitary waves—so‑called “lumps”—that decay algebraically in all spatial directions. Each lump solution can be written as ζₖ(x,y)=−6∂ₓ²log τₖ(x,y), where τₖ is a symmetric polynomial of degree k(k+1). The first two members (k = 1, 2) are explicit; higher‑order lumps have been classified in the literature. Crucially, recent work has shown that these lumps are non‑degenerate: the linearized operator about ζₖ has a trivial kernel when restricted to functions preserving the same even symmetry in x and y. This non‑degeneracy is a key hypothesis for applying an implicit‑function theorem.

The authors consider the FDKP‑I equation  u_t + m(D) u_x + 2uu_x = 0, where the Fourier multiplier m(D) is derived from the exact water‑wave dispersion relation. Expanding m(k) for small wave numbers yields m(k)=˜m(k)+O(|(k₁,k₂k₁)|⁴), where ˜m(k) is the KP‑I symbol. Thus, in a low‑frequency regime the two symbols coincide up to higher‑order corrections.

To construct solitary waves of the steady FDKP‑I equation  −c u + m(D) u + u² = 0, the authors introduce Banach spaces X⊂Z⊂L²(ℝ²) equipped with norms that capture both the anisotropic nature of the problem and the scaling inherent in KP‑I. They split the Fourier space into a low‑frequency cone C (|k₁|≤δ, |k₂k₁|≤δ) and its complement, and decompose any function u into u₁ (spectral support in C) and u₂ (support outside C). The low‑frequency component u₁ is measured with an ε‑dependent norm |·|_ε reflecting the KP scaling (x→εx, y→ε²y). The high‑frequency component u₂ is handled via the operator n(D)=m(D)−I, which is an isomorphism on the complementary subspace.

Using a Lyapunov‑Schmidt‑type reduction, the authors solve the high‑frequency equation for u₂ as a smooth function of u₁. A contraction‑mapping argument yields the estimate ‖u₂‖{X₂} ≲ ε‖u₁‖{X₁}², showing that u₂ is a higher‑order correction. Substituting u₂(u₁) back into the low‑frequency equation leads to a reduced scalar equation for the rescaled profile ζ defined by u₁(x,y)=ε²ζ(εx,ε²y):  ε⁻² n_ε(D) ζ + ζ + χ_ε(D) ζ² + S_ε(ζ) = 0, where n_ε and χ_ε are the ε‑scaled versions of n and the characteristic function of C, and S_ε is a remainder satisfying ‖S_ε(ζ)‖{L²} ≲ ε²‖ζ‖{Y₁}³. As ε→0, χ_ε→I and the equation formally reduces to the KP‑I steady equation ζ + ˜m(D)ζ + ζ² = 0.

At this stage the authors invoke the non‑degeneracy of the KP‑I lumps. For each integer k, the lump ζₖ solves the limiting equation and satisfies the kernel condition required for the implicit‑function theorem. By applying a low‑regularity version of the theorem (to accommodate the ε‑dependent norm), they prove that for sufficiently small ε there exists a unique symmetric solution ζ_{ε,k} of the reduced equation with ζ_{ε,k}→ζₖ in the natural energy space Y₁. Translating back to the original variables yields a family of fully localized solitary waves of the FDKP‑I equation:  u_{k}(x,y) = ε² ζₖ(εx, ε²y) + o(ε²), with wave speed c = 1 − ε². The solutions inherit the even symmetry in both spatial directions and decay algebraically, just as the KP‑I lumps.

The main theorem (Theorem 1.2) thus establishes that the full‑dispersion model retains an infinite hierarchy of fully localized solitary waves, each approximated by a KP‑I lump at leading order. This result bridges the gap between the approximate KP‑I model and the more accurate FDKP‑I model, confirming that the richer dispersion does not destroy the existence of lump‑type solitary structures. The methodology—combining spectral decomposition, Lyapunov‑Schmidt reduction, precise estimates on the remainder, and a non‑degeneracy‑based implicit‑function argument—offers a robust framework for studying solitary waves in other nonlocal dispersive equations where exact dispersion relations are retained. The paper also highlights the physical relevance: in regimes of strong surface tension, three‑dimensional water waves can support highly localized, coherent structures that are mathematically tractable and potentially observable.