Flying Wing Solitons

• Flying wing solitons are multidimensional, directionally localized nonlinear wave structures that form through mechanisms like dispersive shocks and symmetry reductions.
• They manifest across systems such as the nonlinear Schrödinger equation, Bose–Einstein condensates, and Ricci flow, with features determined by parameters like Mach number and cone angle.
• Analytical, numerical, and experimental studies validate their existence and stability, offering insights into soliton interactions, collision dynamics, and geometric singularity models.

Flying wing solitons are multidimensional, directionally localized nonlinear wave structures that emerge in various nonlinear systems, notably dispersive hydrodynamics, nonlinear Schrödinger-type equations, quantum fluids, and geometric flows. The term covers phenomena ranging from oblique dark soliton fans in supersonic superfluid flow to gradient Ricci solitons with sectorial asymptotics in geometric analysis. Central distinguishing features are their pronounced angular localization, controlled by system parameters (e.g., Mach number, cone angle, symmetry), and their formation through mechanisms such as dispersive shocks, quantization rules, or integrating symmetry-reduced ODEs. Flying wing solitons are physically realized as dark lines in Bose–Einstein condensates, directional envelope beams in nonlinear optics and oceanography, self-accelerating matter-wave structures, and geometric singularity models in Ricci flow.

1. Oblique Soliton Fans in Supersonic Nonlinear Schrödinger Flows

Flying wing solitons in the two-dimensional defocusing nonlinear Schrödinger (NLS) equation arise when a superfluid is driven past a slender, impenetrable obstacle at highly supersonic velocity. The governing equations can be asymptotically reduced, via a hypersonic scaling, to a one-dimensional piston-type dispersive problem where the obstacle's spatial profile dictates the "piston" motion curve, transforming the stationary boundary value problem into a time-like evolution equation for the NLS profile (El et al., 2009).

The main structure features:

• Front dispersive shock wave (DSW): Generated at the leading edges of the obstacle. The evolution is analytically captured by @@@@1@@@@ for the periodic NLS solution parameterized by four Riemann invariants $(\lambda_1,...,\lambda_4)$. Edge slopes depend on the local Mach number $M$ and slenderness parameter $\alpha$:
  • Leading edge: $$s^{+} = [2(\alpha M)^2 + 4(\alpha M) + 1]/[M(1+\alpha M)]$$
  • Trailing edge: $s^{-} = 1/M + \alpha/2$
• Rear oblique soliton fan ("flying wing solitons"): In the wake beyond the obstacle, a train of oblique dark solitons is nucleated. Semiclassically, their properties are set by a Bohr–Sommerfeld quantization condition on the initial potential well:
  • Eigenvalues $\lambda_k$ satisfy $\int \sqrt{-\lambda_+(Y,0) + 1} dY = 2\pi(k+1/2)$
  • Soliton amplitude: $a_k = 1 - \lambda_k^2$
  • Slope: $s_k = \lambda_k/M$
  • Each soliton line follows $y_k(x) = (\lambda_k/M)x$
  • The number $N$ of solitons depends on obstacle geometry (e.g., parabolic wings: $$N \sim l (\alpha M)^{1/2}(10+3\alpha M)/(2\pi\sqrt{2})$$, with $l = L/M$)
• Linear ship-wave regime: Beyond the nonlinear DSW zone, linear radiation emerges, forming distinctive wave crests characterized by asymptotic formulas for wavenumber and amplitude decay.
• Stability constraints: The fan structure requires $M \gg 1$, $\alpha \ll 1$, and convective stabilization ($M_l = M/\sqrt{n_p}>1$).

These analytic predictions are validated against direct numerical simulations of the 2D NLS system, with quantitative agreement in soliton amplitude, front edge position, and density dips (El et al., 2009).

2. Integrable and Near-Integrable Oblique Soliton Solutions

Exact single and double flying wing soliton solutions in two-dimensional NLS flows are constructed via Hirota's bilinear method (Khamis et al., 2012). The stationary equation admits soliton solutions at arbitrary angles within the Mach cone, subject to $|M \sin\theta| \leq 1$:

• Single oblique soliton:

$$\psi(x,y,t) = e^{i(\alpha x - \Omega t)} \frac{1 + e^{\xi + 2i\alpha}}{1 + e^{\xi}}$$

with $$\xi = 2\sqrt{1 - (M\sin\theta)^2}(x\sin\theta - y\cos\theta)$$, $$e^{i\alpha} = M\sin\theta + i\sqrt{1 - (M\sin\theta)^2}$$.

• Double soliton ("wing") ansatz: Collisions and interactions of two solitons at distinct angles $\theta_1, \theta_2$ introduce an interaction constant $\phi_{12}$, yielding "near-integrable" dynamics—collision outcomes are elastic except for sufficiently large relative angles or moderate $M$.
• Mach cone constraint: Oblique solitons exist only within $|\theta| < \arcsin(1/M)$.

Physical interpretation confirms that a supersonic obstacle nucleates pairs of oblique dark solitons at finite angle, each exhibiting convective stability. These structures reproduce key characteristics of the rear soliton fan in the dispersive piston regime and admit nearly elastic collisions (Khamis et al., 2012).

3. Directional Flying Wing Solitons and Breathers in (2D+1) NLSE

In systems governed by the (2D+1) hyperbolic nonlinear Schrödinger equation, directional or "slanted" envelope solitons and breathers propagate at an angle $\theta$ to the carrier wave crests, forming coherent beams with finite transverse crest length—"flying wing beams"—across hydrodynamics, optics, BECs, and plasma (Chabchoub et al., 2018):

• Coordinate transformation: Introducing $T = t\cos\theta - (y/c_g)\sin\theta$ reduces the original 2D+1 NLSE to a 1D+1 focusing NLSE for $\psi(x,T)$, permitting analytic soliton and Peregrine breather solutions.
• Existence condition: Localized beams exist for $|\theta| < \arcsin(1/\sqrt{3}) \approx 35.26^\circ$, set by the sign of the group-velocity dispersion $\Lambda$.
• Analytical solution for envelope beam:

$$\psi(x,T) = A\, \mathrm{sech}[A\sqrt{\Gamma/\Lambda}(T - T_0 - x/C_g)] \exp\{ i [(\Gamma A^2/2)x - \Omega T + \phi_0 ] \}$$

• Crest length in the $y$ direction: $\Delta y = 2c_g/[A\sqrt{\Gamma/\Lambda}\sin\theta]$
  • Experimental confirmation: Water wave basin experiments reconstruct the full 3D evolution via marker nets and stereo imaging, matching theoretical predictions within 5% uncertainty (Chabchoub et al., 2018).

This directional localization mechanism underpins finite-length rogue waves, directional soliton beams in Kerr media, and slanted transport in matter-wave systems, linking experimental realizations with the analytical structure of flying wing solitons.

4. Flying Wing Solitons in Quantum Fluids Beyond Mean Field

In Lee–Huang–Yang quantum fluids—where the nonlinear term in the governing equation is cubic-quintic—flying wing solitons remain robustly supported under supersonic flow past obstacles (Santos et al., 10 May 2025). The soliton profile is constructed by embedding a 1D solution into the lab frame, allowing for direct mapping of Mach number to soliton angle:

• Soliton angle-Mach relation: $\theta = \arccos(1/M)$ denotes the interior of the Mach cone, dictating admissible soliton orientations.
• Analytical profile: The density profile is implicitly given via

$$x\cos\theta + y\sin\theta = \pm \frac{1}{2} \int_{\rho_m}^{\rho} \frac{d\rho}{\sqrt{R(\rho)}}$$

with $R(\rho)$ determined by cubic-quintic nonlinearity.

• Amplitude and phase jump: The soliton depth and phase discontinuity follow from double-root analysis, paralleling the classical GPE structure but modified quantitatively by the higher-order nonlinearity.
• Emergence and stability: Numerical simulations reveal stable soliton formation for $M \gtrsim 3$, with lower threshold leading to snake instability; profiles match analytical predictions.

This demonstrably extends flying wing soliton physics to quantum fluids with nonstandard nonlinearities, confirming the universal character of oblique soliton formation mechanisms (Santos et al., 10 May 2025).

5. Geometric Flying Wings: Steady Gradient Ricci Solitons

Flying wing solitons in geometric analysis comprise steady gradient Ricci soliton manifolds with asymptotic cone shaped as a sector of prescribed angle. In dimension $n=3$, the seminal construction of Lai and others provides a one-parameter family of noncompact solitons called "flying wings" (Lai, 2022, Lai, 2020, Lai, 2022):

• Definition: A steady Ricci soliton $(M^n,g,f)$ satisfies $\operatorname{Ric} + \nabla^2 f = 0$, with $f$ the potential function, attaining a unique critical point (tip). The metric is $O(2)$-symmetric with a single geodesic axis, and, in $3$D, $M$ is asymptotic to a sector of opening $\theta$.
• ODE reduction: The metric is constructed via warped-product ansatz and reduces to coupled ODEs for the warping function and potential; see

$$g = ds^2 + a(s)^2 d\alpha^2 + b(s)^2 d\theta^2, \quad s \in \mathbb{R}$$

with $a(s), b(s), f(s)$ subject to tip and infinity boundary conditions.

• Classification: All $3$D steady Ricci solitons of positive curvature are either:
  • Bryant soliton (rotationally symmetric, asymptotic to a ray),
  • Flying wing (nonrotational, $O(2)$ symmetric, asymptotic to a sector).
• Curvature properties: Scalar curvature remains positive at infinity, converging to $\sin^2(\theta/2)$, and sectional curvatures are nonnegative. Volume growth is subcubic ("collapsed") in dimension $3$; higher-dimensional analogues are noncollapsed.
• Higher-dimensional generalization: For $n \geq 4$, steady Ricci solitons with prescribed tip Ricci eigenvalues yield an $(n-2)$-parameter family of flying wings, with noncollapsing when additional symmetries are imposed (Chan et al., 27 Oct 2025). The construction leverages Ricci flow smoothing from spherical polyhedral links via scaling-invariant estimates.

6. Kähler Flying Wings: Non-rotationally Symmetric Steady Kähler–Ricci Solitons

In complex geometry, flying wing solitons manifest as $U(1) \times U(n-1)$-invariant, but non-$U(n)$-invariant, steady gradient Kähler–Ricci solitons on $\mathbb{C}^n$, providing the Kähler counterpart to Riemannian flying wings (Chan et al., 2024). These structures interpolate between the fully symmetric Cao soliton and a product of a 2D cigar and a lower-dimensional Cao soliton.

• Metric ansatz: On $\mathbb{C}^n$, the metric is

$$g = dr^2 + (b(r))^2 g^T + (a(r))^2 \eta^2, \quad a(r) = b(r) b'(r)$$

where $g^T$ is the Fubini–Study metric and $\eta$ the Hopf contact form. Kählerity is ensured by the $a = b b'$ condition.

• ODE reduction: The soliton equation reduces to a single ODE for $b(r)$, with regularity enforced at the tip ($r=0$) and prescribed asymptotics at infinity.
• Curvature operator: Sectional curvature is strictly positive on real $(1,1)$-forms; explicit scalar inequalities on $a,b$ and their derivatives govern curvature positivity.
• Asymptotic cone: As $r \to \infty$, the metric approaches a cone with link metric capturing the flying wing sector. The deformation parameter $\alpha$ indexes the family, connecting the symmetric endpoint to cigar-product geometry.
• Intermediate results: The construction incorporates Sasaki-to-cone lift, continuity methods for expanders, coarea/volume identities, and explicit curvature estimates.

Kähler flying wings answer the uniqueness question for steady Kähler–Ricci solitons negatively in $n \geq 3$, introduce new moduli of wing-shaped steady solitons, and manifest geometric rigidity failure even under strong curvature lower bounds (Chan et al., 2024).

7. Self-Accelerating "Flying Wing" Solitons in Nonlocal Nonlinear Media

In models for two-component Bose–Einstein condensates with microwave-mediated long-range intercomponent interactions, robust self-accelerating, tail-free 2D solitons and vortices—a class of "flying wing" solitons—are generated via transformation of stationary solutions with arbitrary acceleration parameters (Qin et al., 2019):

• GPE+Poisson system:

$$i\partial_t \Psi = [ -\frac{1}{2}\nabla^2 \Psi + \text{SOC/Rabi terms} - \beta|\Psi|^2 \Psi - 2H(x,y,t) \Psi ]$$

$\nabla^2 H = -\gamma |\Psi|^2$

The nonlocal MW feedback yields an effective trapping potential (linear in 1D, logarithmic in 2D), ensuring tail-free localization.

• Acceleration via Galilean-type transformation:

$x' = x - V_x t - \frac{1}{2} a_x t^2$

$$\Psi(x,y,t) = \Psi'(x',y',t) \exp[i(\alpha_x(t)x+\alpha_y(t)y-\phi(t))]$$

The soliton center-of-mass obeys $x_c(t) = V_x t + \frac{1}{2} a_x t^2$, permitting arbitrary self-accelerating trajectories.

• Stability under interactions/SOC: Solitons and vortices remain stable under weak cubic nonlinearity and SOC perturbation, and can be realized optically in thermal media by analogous nonlocality-induced mechanisms.

This soliton family demonstrates a mechanism for "flying wing" self-accelerating localization, extending the taxonomy of flying wing structures beyond stationary hydrodynamic and geometric contexts (Qin et al., 2019).

---

Flying wing solitons thus constitute a unifying concept encompassing oblique quantized soliton fans in dispersive hydrodynamics, integrable slanted envelope solutions in multidimensional nonlinear Schrödinger systems, self-accelerating tail-free matter-wave and optical solitons, as well as sectorial singularity models in Ricci geometric flow. Their analytic construction, experimental realization, and geometric classification continue to elucidate the deep interrelations between symmetry, nonlinear localization, and multidimensional geometry in nonlinear science.