Spectral stability of finite-amplitude interfacial solitary waves: oscillatory disturbances, infinite depth, the overhanging limiting regime, and the transverse remainder

For finite-amplitude interfacial gravity solitary waves in a two-layer fluid of finite depth, Kataoka (Fluid Dynamics Research, 2006) proved that exchanges of stability against disturbances that are stationary relative to the wave occur at every stationary value of the total wave energy along a non-bifurcating solution branch, and derived the small growth rates near those critical amplitudes. That analysis left the stability classification incomplete in four directions: eigenvalues with nonzero imaginary part (nonstationary, oscillatory disturbances) were not treated; the infinite-depth case, where solitary-wave tails decay algebraically and the exponential-tail asymptotics fail, was not covered; three-dimensional (crestwise, transverse) stability was deferred; and the stability boundary near the overhanging limiting configurations was left numerically unresolved. Later work resolved the long-wavelength transverse question (Kataoka, JFM, 2008) and computed near-limiting overhanging configurations (Maklakov & Sharipov 2018; Guan et al. 2021), but the oscillatory-eigenvalue problem, the infinite-depth extension, the transverse spectrum at finite crestwise wavenumbers, and the stability map in the overhanging regime remain untreated. The problem posed here is to complete this spectral stability classification for the two-layer Euler system.

Background

When two immiscible, inviscid, incompressible fluids of densities $\rho_{\mathrm{U}}<\rho_{\mathrm{L}}$ are stacked between rigid horizontal lids, the sharp interface between them can support interfacial (internal) solitary waves - localized humps of the interface travelling at constant speed $v$ above the linear long-wave speed. In nondimensional form the system is characterized by the density ratio $\rho=\rho_{\mathrm{U}}/\rho_{\mathrm{L}}$ and the depth ratio $D=D_{\mathrm{U}}/D_{\mathrm{L}}$ (upper over lower layer); interfacial tension is neglected. Finite-amplitude solution branches of this two-layer Euler problem were computed by Pullin & Grimshaw (1988), and for sufficiently large amplitude portions of the interface can become overhanging, i.e. the interface doubles back so that it is no longer a graph, a geometry first computed for periodic interfacial waves by Meiron & Saffman (1983).

Linear (spectral) stability is decided by the eigenvalue problem for infinitesimal disturbances proportional to $e^{\lambda t}$ in the frame moving with the wave; $\mathrm{Re}\,\lambda>0$ means growth. Disturbances that are stationary relative to the basic wave correspond to the superharmonic class familiar from periodic-wave theory (disturbances with the same spatial scale as the wave itself, as opposed to slowly modulated subharmonic disturbances). An exchange of stability is the passage of a real eigenvalue through $\lambda=0$ as the wave amplitude varies, creating or destroying a growing mode. For surface gravity waves on a single fluid, energy-extremum criteria for such exchanges had been developed numerically and asymptotically, including by Kataoka (2006, JFM) for waves of finite depth, and Lin (2008/2009) later proved rigorously that surface solitary waves become linearly unstable beyond the wave of maximal energy. For small amplitudes, interfacial solitary waves are stable within Korteweg–de Vries (KdV) type long-wave model equations - the weakly nonlinear, weakly dispersive approximations for long interfacial waves.

Kataoka (2006, Fluid Dynamics Research) extended the finite-amplitude theory to interfacial solitary waves: performing an asymptotic analysis for small growth rates of stationary disturbances, he showed that an exchange of stability occurs at every stationary value of the total energy $E$ as a function of wave speed $v$ (i.e. where $\mathrm{d}E/\mathrm{d}v=0$ along a simple solution branch), obtained an explicit asymptotic formula for the small growth rate near the critical amplitude whose sign also fixes whether the number of growing modes increases or decreases, and verified the predictions against numerical eigenvalue computations. Applied to computed branches, this revealed features absent in the single-layer surface case: for elevation waves (humps of the interface into the upper layer) a critical density ratio $\rho_{\mathrm{cr}}(D)$ below which instabilities and even restabilizations (multiple energy extrema along one branch) occur, monotone energy and no detected exchange for the depression cases (dips of the interface into the lower layer) examined, and the onset of overhanging interface geometry at amplitudes that need not coincide with the stability threshold.

The 2006 analysis is restricted in three structural ways: it treats only stationary disturbances (real eigenvalues near zero), only finite-depth configurations (the asymptotics rely on the exponential decay of the solitary-wave tails, with the far field scaled by $|\lambda|^{-1}$), and only two-dimensional disturbances. The transverse (crestwise) stability question - disturbances varying along the wave crest, the direction perpendicular to propagation - was answered in the long-wavelength regime by Kataoka (2008), who derived an explicit criterion and proved the general statement that the solution becomes transversely unstable before an exchange of longitudinal stability occurs; the analogous surface-wave transverse theory is due to Kataoka & Tsutahara (2004). The remaining gaps - oscillatory eigenvalues, infinite depth, the overhanging limiting regime, and the transverse spectrum at finite crestwise wavenumbers - define the problem below.

Problem Statement

For the two-layer Euler system - two immiscible, inviscid, incompressible homogeneous fluids of densities $\rho_{\mathrm{U}}<\rho_{\mathrm{L}}$ between rigid horizontal lids (in the infinite-depth configurations of item 2, the lid bounding that layer is removed and the layer is unbounded), with a sharp interface, no interfacial tension, and density and depth ratios $\rho$ and $D$ - complete the spectral stability classification of finite-amplitude interfacial solitary waves initiated by Kataoka (2006), by resolving the linked questions that work left open:

1. Oscillatory disturbances. Analyze the linearized eigenvalue problem for nonstationary disturbances, i.e. eigenvalues $\lambda$ with $\mathrm{Im}\,\lambda\neq 0$: determine, over the full admissible $(\rho, D)$ range (or over a sub-range whose complement is excluded by explicit argument), whether unstable oscillatory modes exist along the elevation and depression solitary-wave branches (in particular near and between the energy-extremum exchanges of stability), exhibiting an unstable oscillatory mode or establishing that none exists for each configuration, so that the stability classification no longer restricts to real eigenvalues near zero.

2. Infinite depth. Extend the exchange-of-stability analysis to configurations in which the upper or lower layer is infinitely deep, where the solitary-wave tails decay algebraically rather than exponentially and the finite-depth small-$|\lambda|$ asymptotics (with far field scaled by $|\lambda|^{-1}$) do not apply: over the full admissible range of the remaining parameters (or over a sub-range whose complement is excluded by explicit argument), either derive the analogous growth-rate asymptotics and energy-extremum criterion there, or prove that the finite-depth energy-extremum criterion does not carry over and exhibit explicitly the condition that governs exchanges of stability in the infinite-depth limit.

3. Overhanging limiting regime. Determine, analytically or numerically, over the full admissible $(\rho, D)$ range (or over a sub-range whose complement is excluded by explicit argument), the precise limiting amplitudes and the boundary between unstable solution branches and the no-solution regions for overhanging interfacial solitary waves - i.e. how the stationary-disturbance stability map in the $(\rho, D)$ plane terminates near the limiting configurations.

4. Transverse remainder. Complete the three-dimensional stability picture beyond the long-wavelength regime resolved by Kataoka (2008), i.e. determine the transverse spectrum at finite crestwise wavenumbers (the transverse growth rate as a function of crestwise wavenumber) across the finite-wavenumber range left open by Kataoka (2008) and over the full admissible $(\rho, D)$ range (or over a sub-range whose complement is excluded by explicit argument), identifying any band of unstable wavenumbers or establishing its absence.

These four components jointly serve one objective - a complete spectral stability map of finite-amplitude interfacial solitary waves across depth configurations and disturbance classes. All four are required: resolving only a proper subset of items 1-4 does not complete the classification.

References

1. Takeshi Kataoka (2006), 'The stability of finite-amplitude interfacial solitary waves', Fluid Dynamics Research 38(12), 831–867. DOI: 10.1016/j.fluiddyn.2006.02.008, https://doi.org/10.1016/j.fluiddyn.2006.02.008
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3. Takeshi Kataoka (2013), 'The superharmonic instability of finite-amplitude interfacial waves', conference proceedings.
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