Full-Dispersion KP-I Equation

• The full-dispersion KP-I equation is a nonlinear dispersive model that incorporates the exact water-wave dispersion relation for enhanced gravity–capillary wave analysis.
• It employs nonlocal operators and precise Fourier symbols to characterize two-dimensional lump solitary waves with algebraic decay.
• Improved dispersive estimates and well-posedness results lower the regularity threshold compared to classical KP-I, extending its use to elastic wave regimes.

The full-dispersion KP-I equation (FDKP-I) constitutes a nonlinear dispersive evolution model that retains the exact linear dispersion relation from the underlying water-wave problem, rather than a truncated low-frequency Taylor expansion. In the regime of strong surface tension (Bond number $\beta>1/3$), FDKP-I emerges as an extension of the Kadomtsev–Petviashvili I (KP-I) equation for gravity–capillary waves, rectifying limitations within classical models and permitting a precise characterization of two-dimensional "lump" solitary waves with algebraic decay (Ehrnström et al., 17 Dec 2025, Ehrnström et al., 2018, Pilod et al., 2020). Formally, FDKP-I appears either as a PDE with nonlocal operator $m(D)$ acting in physical space or, equivalently, via the Fourier symbol $m(k_1,k_2)$ inherited from the full Euler water-wave dispersion. It has recently been explored for both water-wave and elastic-wave contexts (Erbay et al., 2022).

1. Mathematical Formulations and Dispersion Symbols

In nondimensional variables (with $g=1$, $d=1$, $\rho=1$), and for surface tension $\sigma$ represented via the Bond number $\beta = \frac{\sigma}{\rho g d^2}$, FDKP-I is formulated as: $u_t + m(D) u_x + 2 u u_x = 0,$ where $D = -i(\partial_x, \partial_y)$ is the Fourier differentiation operator. The nonlocal operator $m(D)$0 possesses a symbol defined as: $m(D)$1 This symbol reflects the precise phase velocity for linear water waves. In physical coordinates, a gravity-capillary FDKP-I model appears as: $m(D)$2 with

$m(D)$3

In the context of dispersive elastic waves, analogous FDKP-type equations utilize nonlocal operators $m(D)$4 built from elasticity kernel transforms, indicating the generality of the full-dispersion equation paradigm (Erbay et al., 2022).

2. Derivation from Water-Wave and Elastic Wave Models

FDKP-I arises from the full three-dimensional, irrotational, incompressible Euler equations for water waves with strong capillarity, where the dispersion relation for plane waves is

$m(D)$5

For $m(D)$6, the phase-velocity function $m(D)$7 attains a unique global minimum at $m(D)$8. Solitary waves bifurcate at near-minimum speed $m(D)$9, justified by the governing dispersive structure. Classical KP-I emerges from weakly-dispersive expansions of $m(k_1,k_2)$0: $m(k_1,k_2)$1 whereas FDKP-I retains the full symbol, enhancing physical fidelity for solitary wave phenomena (Ehrnström et al., 17 Dec 2025, Ehrnström et al., 2018). In similar fashion, elastic wave analogues employ full-dispersion operators based on nonlocal elasticity kernels to describe long, small-amplitude anti-plane shear waves (Erbay et al., 2022).

3. Lump Solitary Waves: Existence and Structure

Both KP-I and FDKP-I equations admit algebraically localized solitary wave solutions—known as "lumps"—in the strong surface tension regime. For classical KP-I, explicit rational lump solutions are constructed as: $m(k_1,k_2)$2 where $m(k_1,k_2)$3 is a symmetric real polynomial of total degree $m(k_1,k_2)$4. The FDKP-I equation admits fully localized solitary waves constructed as perturbative deformations of the classical lumps, i.e.,

$m(k_1,k_2)$5

with amplitude parameter $m(k_1,k_2)$6 as $m(k_1,k_2)$7 (the bifurcation speed). The lump solutions are smooth ($m(k_1,k_2)$8) and exhibit algebraic decay, with

$m(k_1,k_2)$9

A family of such lumps exists, indexed by the lump number $g=1$0 and inheriting symmetries from classical solutions. In the FDKP-I context, lump existence is established using perturbative Lyapunov–Schmidt reduction, low-/high-frequency decomposition, and application of an implicit-function theorem based on the nondegeneracy of classical KP-I lumps (Ehrnström et al., 17 Dec 2025, Ehrnström et al., 2018).

4. Dispersive and Strichartz Estimates; Well-posedness

FDKP-I exhibits substantial improvements over the classical KP-I in dispersive regularity and well-posedness. The localised $g=1$1 decay of the linear solution operator is established as: $g=1$2 where $g=1$3 is a Littlewood–Paley frequency projector and $g=1$4; the decay is proven using stationary phase and sharp asymptotics for asymmetric Bessel functions (Pilod et al., 2020). Strichartz estimates of the form

$g=1$5

are derived via $g=1$6 and Hardy–Littlewood–Sobolev theory.

These dispersive bounds allow for local well-posedness of the nonlinear initial-value problem in the capillary–gravity regime for data in $g=1$7, for $g=1$8: $g=1$9 with flow map continuity and uniqueness. The regularity threshold is lowered below the classical $d=1$0 due to two-dimensional dispersive effects not present in KP-I. For FDKP-I, no "zero-mass constraint" arises and the group is unitary in all $d=1$1 (Pilod et al., 2020).

5. Comparison: Classical KP-I Versus Full-Dispersion KP-I

Classical KP-I is characterized by a dispersion symbol $d=1$2, which is singular at $d=1$3, necessitating zero-mass constraints and resulting in insufficient regularization at low frequencies. FDKP-I replaces this by a bounded and smooth nonlocal symbol, avoiding mass constraints and improving low-frequency regularity. Nonlinearity in both models remains quadratic. Lump solitary waves in KP-I exist for arbitrary amplitude; in FDKP-I, the lumps persist for sufficiently small amplitude and better approximate the true water-wave solutions by virtue of exact dispersion (Ehrnström et al., 17 Dec 2025, Ehrnström et al., 2018).

The FDKP-I lumps converge uniformly to classical KP-I lumps under the scaling $d=1$4 as amplitude $d=1$5. Thus, FDKP-I bridges the gap between weakly-dispersive KP-I and the complete water-wave problem (Ehrnström et al., 17 Dec 2025).

6. Extensions: Full-Dispersion KP Equations in Elastic Media

Generalizations of the full-dispersion KP framework appear in nonlinear elasticity, particularly for anti-plane shear waves in nonlocal elastic media. Two principal models are developed: the Whitham-type full-dispersion KP-I equation and the BBM-type full-dispersion KP equation. In the Whitham-type, the strain variable $d=1$6 satisfies: $d=1$7 with $d=1$8 constructed from the elasticity kernel via its Fourier transform. The BBM-type modifies the time derivative's dispersive weight. Both models recover classical KP-I behavior in the long-wave limit and admit further simplified forms via operator expansions (Erbay et al., 2022). For the Whitham-type equation, line solitary waves are subject to transverse instability when the propagation speed exceeds a critical value ($d=1$9), as shown by spectral analysis.

7. Variational and Analytical Techniques

FDKP-I solitary wave existence theory blends variational principles, finite-dimensional reduction (bow-tie region in phase-space), and perturbative analysis. Solitary waves are identified as constrained critical points of the energy functional: $\rho=1$0 subject to fixed momentum. Natural constraint sets and anisotropic functional spaces are leveraged, and existence is established through minimisation arguments, Ekeland’s variational principle, and concentration–compactness methods. The nondegeneracy of KP-I lumps and Fredholm theory are critical in establishing the uniqueness and stability of solutions (Ehrnström et al., 2018, Ehrnström et al., 17 Dec 2025).

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In summary, the full-dispersion KP-I equation is a physically precise nonlinear dispersive model for two-dimensional gravity–capillary water waves (and analogous elastic waves) in the strong surface tension regime. By accurately reflecting the full dispersion relation, it supports a family of localized lump solitary waves, improves dispersive regularization, facilitates analytical well-posedness at lower regularity, and corrects several artifacts of classical KP-I. Extensions to elasticity further establish FDKP-type equations as a universal paradigm for modeling fully dispersive long-wave phenomena.