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 $m(D)$ possesses a symbol defined as: $$m(k_1, k_2) = \frac{\omega(k)}{k_1}, \qquad \omega(k) = \sqrt{k (1+\beta k^2) \tanh k},\quad k = \sqrt{k_1^2 + k_2^2}\,.$$ This symbol reflects the precise phase velocity for linear water waves. In physical coordinates, a gravity-capillary FDKP-I model appears as: $$\partial_t u + \widetilde L_{\beta}(D) u + 3\,\partial_{x_1}(u^2) = 0,$$ with

$$\widetilde L_{\beta}(D) = \frac{i D_1 |D_1| |D| (1+\beta|D|^2)^{1/2} (\frac{\tanh |D|}{|D|})^{1/2}}, \quad |D| = \sqrt{D_1^2 + D_2^2}.$$

In the context of dispersive elastic waves, analogous FDKP-type equations utilize nonlocal operators $L(D_x, D_y)$ 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

$\omega^2(k) = k (1+\beta k^2) \tanh k.$

For $\beta > 1/3$, the phase-velocity function $c(k_1) = \omega(k)/k_1$ attains a unique global minimum at $k_1 = 0$. Solitary waves bifurcate at near-minimum speed $c \approx 1$, justified by the governing dispersive structure. Classical KP-I emerges from weakly-dispersive expansions of $m(k_1,k_2)$: $$m(k_1, k_2) = 1 + \frac{1}{2} (\beta - {\textstyle\frac{1}{3}}) k_1^2 + \frac{k_2^2}{k_1^2} + O(|k_1|^4, (k_2/k_1)^4),$$ 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: $$\zeta_k^\star(x, y) = -6 \,\partial_x^2 \log \tau_k^\star(x, y),$$ where $\tau_k^\star(x, y)$ is a symmetric real polynomial of total degree $k(k+1)$. The FDKP-I equation admits fully localized solitary waves constructed as perturbative deformations of the classical lumps, i.e.,

$$u_k^\star(x, y) = \varepsilon^2\,\zeta_k^\star(\varepsilon x, \varepsilon^2 y) + o(\varepsilon^2),$$

with amplitude parameter $\varepsilon = \sqrt{1-c}$ as $c \to 1^-$ (the bifurcation speed). The lump solutions are smooth ($H^\infty(\mathbb{R}^2)$) and exhibit algebraic decay, with

$$|\partial_x^{m_1}\partial_y^{m_2}u_k^\star(x,y)| \lesssim \frac{\varepsilon^{2+m_1+2m_2}}{(1+\varepsilon^2 x^2 + \varepsilon^4 y^2)^{1+(m_1+m_2)}}.$$

A family of such lumps exists, indexed by the lump number $k$ 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 $L^1 \to L^\infty$ decay of the linear solution operator is established as: $$\|\exp[t L] P_\Lambda f\|_{L_x^\infty} \le c_\beta [\sqrt{\beta}\Lambda]^{-1} \Lambda^{3/2} |t|^{-1} \|P_\Lambda f\|_{L_x^1},$$ where $P_\Lambda$ is a Littlewood–Paley frequency projector and $c_\beta > 0$; the decay is proven using stationary phase and sharp asymptotics for asymmetric Bessel functions (Pilod et al., 2020). Strichartz estimates of the form

$$\|\exp[t L] P_\Lambda f\|_{L_t^q L_x^r} \le c_\beta [\sqrt{\beta}\Lambda]^{-(1/2-1/r)} \Lambda^{3/2(1/2-1/r)}\|P_\Lambda f\|_{L_x^2}$$

are derived via $TT^*$ 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 $H^s(\mathbb{R}^2)$, for $s > 7/4$: $$u \in C([0,T]; H^s(\mathbb{R}^2)) \cap L^1((0,T); W^{1,\infty}(\mathbb{R}^2)),$$ with flow map continuity and uniqueness. The regularity threshold is lowered below the classical $s>2$ 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 $H^s$ (Pilod et al., 2020).

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

Classical KP-I is characterized by a dispersion symbol $k^3 - \ell^2/k$, which is singular at $k=0$, 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 $x \to \varepsilon x, y \to \varepsilon^2 y$ as amplitude $\varepsilon \to 0$. 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 $v$ satisfies: $v_t + L(D_x, D_y) v_x + \mu v^2 v_x = 0,$ with $L(D_x, D_y)$ 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 ($c>4$), 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: $$\mathcal{E}(u) = \tfrac{1}{2}\int_{\mathbb{R}^2}|(m(D))^{1/2} u|^2\,dx\,dy + \tfrac{1}{3}\int_{\mathbb{R}^2}u^3\,dx\,dy,$$ 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).

---

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.