Kim (2009) showed by formal perturbation theory within the full two-layer Euler equations that interfacial gravity–capillary plane solitary waves in deep water — wavepacket-type solitary waves bifurcating just below the minimum interfacial phase speed when interfacial tension acts between two inviscid, immiscible, nearly equally dense fluids — are linearly unstable to long-wavelength transverse disturbances, with only the leading-order growth-rate coefficient determined and the instability supported by an energetic criterion. Subsequent literature keeps the three-dimensional theory of these waves open at the level of the complete Euler problem: rigorous results cover only the one-layer free-surface problem, and no rigorous proof of transverse instability exists for the two-layer interfacial configuration. Related questions — a construction, rigorous or fully nonlinear computational, of fully localized three-dimensional interfacial gravity–capillary solitary waves (lumps) beyond weakly nonlinear model equations, and the computation of the bandwidth of unstable transverse wavenumbers and higher-order growth-rate corrections — also remain open; they are context for this problem, not deliverables of it. The problem asks to upgrade Kim (2009)'s formal perturbation analysis and energetic sign check to a rigorous linear-instability theorem for the full two-layer deep-water Euler equations.

Background

Interfacial (internal) waves propagate along the interface between two immiscible inviscid fluids, a lighter one of density ρ1\rho_1 above a heavier one of density ρ2\rho_2. In the two-layer potential-flow idealization each layer is irrotational, so the velocity in each layer is the gradient of a potential satisfying Laplace's equation; the interface is advected by the flow (kinematic condition) and the pressure jump across it is fixed by interfacial tension TT (dynamic condition). When both layers are infinitely deep (the deep-water configuration of the source), the linear dispersion relation is ω2=(ρ2−ρ1)gk+Tk3ρ1+ρ2,\omega^2=\frac{(\rho_2-\rho_1)gk+Tk^3}{\rho_1+\rho_2}, with gg gravity, kk the wavenumber and ω\omega the frequency. Unlike the pure-gravity case, the phase speed c(k)=ω/kc(k)=\omega/k attains a strict minimum at a finite wavenumber where gravity and capillarity balance, and — as first shown by Benjamin (1967) for internal waves and Benjamin (1992) for the free surface — weakly nonlinear solitary waves bifurcate at speeds just below this minimum: they are modulated wavepackets with carrier wavenumber near the minimum, having oscillatory decaying tails (Dias and Iooss 1993). These are the interfacial gravity–capillary (GC) solitary waves in question.

A plane solitary wave is localized in the direction of propagation and uniform in the transverse horizontal direction. A transverse (cross-wave) instability is a linear instability to disturbances varying in the transverse direction; physically it makes a two-dimensional wave break up into a train of lumps, solitary waves fully localized in both horizontal directions. Nonlinear transverse instability is the stronger form of the transverse instability. For Hamiltonian evolution equations (evolution equations conserving a mechanical energy), such as the Euler equations, an energetic criterion links the two: if the total mechanical energy of a solitary wave decreases as its speed increases, conservation of transverse momentum forces a pair of eigenvalues with positive real part for long transverse perturbations (Bridges 2001; the criterion was made rigorous by Rousset and Tzvetkov in water-wave and model settings). Within a weakly nonlinear model for interfacial GC waves in deep water — the two-dimensional Benjamin equation, a weakly nonlinear bidirectional equation retaining the full interfacial dispersion relation — Kim and Akylas (2006) verified this criterion and computed the resulting lumps. Kim (2009) then carried the analysis into the full two-layer Euler equations: by a formal perturbation expansion in the weakly nonlinear, nearly-equal-density regime he verified the energetic criterion and obtained the leading-order growth rate of long-wave transverse perturbations, establishing the instability at asymptotic order but not rigorously.

The rigorous three-dimensional theory is well developed for the one-layer free-surface analog: fully localized gravity–capillary solitary waves have been constructed mathematically for water of finite depth (Groves and Sun 2008; Buffoni, Groves, Sun and Wahlén 2013) and of infinite depth (Buffoni, Groves and Wahlén 2022), and the transverse instability of line solitary water waves has been proved rigorously (Rousset and Tzvetkov 2010), with continuing numerical and experimental work on the surface case (Park and Cho 2017; Abid, Kharif, Hsu and Chen 2019). For the two-layer interfacial problem with interfacial tension, however, only weakly nonlinear model-equation numerics (Kim and Akylas 2006; Wang, Vanden-Broeck and Meng 2014; Li and Wang 2021 — the last confirming numerically that plane interfacial GC solitary waves are transversely unstable and evolve into lumps) and formal asymptotics are available; moreover, the weakly nonlinear nonlinear-Schrödinger (NLS) envelope approximation for interfacial GC waves changes type at a critical density ratio (Wang, Vanden-Broeck and Meng 2014), so the existence and stability picture may differ qualitatively between density-ratio regimes. The open problem is to bring the two-layer interfacial problem, at the level of transverse instability, to the same rigorous standard as its one-layer counterpart. As a background note on the source's generality, Kim (2009) remarks that its analysis is also applicable to other interfacial gravity–capillary solitary waves that may bifurcate below the minimum of the phase speed, regardless of any restrictions on fluid depths in two-layer potential flows; the target scope of the present problem remains the deep-water configuration.

Problem Statement

Consider the two-layer Euler problem of the source: two inviscid, incompressible, immiscible fluids of densities ρ1<ρ2\rho_1<\rho_2 separated by an interface carrying interfacial tension, in the deep-water configuration (both layers infinitely deep). Establish the rigorous instability status of the interfacial gravity–capillary plane solitary waves that bifurcate at speeds just below the minimum interfacial phase speed: prove — or disprove — as a theorem for the full two-layer Euler equations that these plane solitary waves — the wavepacket solitary waves bifurcating just below the minimum interfacial phase speed in the deep-water configuration, in the weakly nonlinear, nearly-equal-density regime of the source — are linearly unstable to long-wavelength transverse perturbations, upgrading the formal perturbation analysis and energetic sign check of Kim (2009) to a rigorous result. Establish a rigorous instability result for this family, covering a locally constructed near-bifurcation branch of these solitary waves.

Results established only within weakly nonlinear model equations, such as the two-dimensional Benjamin equation studied by Kim and Akylas (2006), Wang, Vanden-Broeck and Meng (2014) and Li and Wang (2021), resolve parts of the mechanism but do not complete the problem as stated for the Euler equations.

References

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