[Paper Review] Heteroclinic orbits, mobility parameters and stability for thin film type equations

Richard S. Laugesen, M. C. Pugh|ArXiv.org|Mar 29, 2000

Fluid Dynamics and Thin Films11 references3 citations

TL;DR

This paper numerically investigates heteroclinic orbits, stability, and the influence of mobility parameters in a thin film equation with nonlinear fourth- and second-order diffusion terms. Using energy-based analysis and numerical simulations, it demonstrates that perturbations of unstable periodic steady states evolve toward either constant or droplet-like steady states, with the long-term behavior dictated by energy minimization, and shows that heteroclinic connections persist under mobility parameter changes despite altered timescales.

ABSTRACT

We study numerically the phase space of the evolution equation h_t = -(h^n h_{xxx})_x - B (h^m h_x)_x . Here h(x,t) is nonnegative, n>0 and m is real, and the Bond number B is positive. We pursue three goals: to investigate the nonlinear stability of the positive periodic and constant steady states; to locate heteroclinic connecting orbits between these steady states and the compactly supported 'droplet' steady states; and to determine how these orbits change when the 'mobility' exponents n and m are changed. For example, we change the mobility coefficients in such a way that the steady states are unchanged and find evidence that heteroclinic orbits between steady states are perturbed but not broken. We also find that when there appear to be touch-down singularities, the exponent n affects whether they occur in finite or infinite time. It also can affect whether there is one touch-down or two touch-downs per period.

Motivation & Objective

• To investigate the nonlinear stability of constant and periodic steady states in a thin film equation with power-law mobilities.
• To locate and analyze heteroclinic orbits connecting periodic, constant, and compactly supported droplet steady states.
• To determine how changes in mobility exponents $n$ and $m$ affect solution dynamics, including finite-time singularities and stability.
• To assess the persistence of heteroclinic connections under variations in mobility parameters while preserving steady states and energy levels.
• To explore critical mobility exponents that govern the onset of finite-time touch-down singularities.

Proposed method

• Numerical continuation and simulation of the thin film equation $h_t = -(h^n h_{xxx})_x - \mathcal{B}(h^m h_x)_x$ on a periodic domain with adaptive time-stepping.
• Energy-based analysis using the gradient flow structure of the equation with respect to a time-dependent weighted $H^{-1}$ inner product.
• Bifurcation diagrams and weakly nonlinear analysis to classify stability of steady states based on $m-n$.
• Systematic variation of mobility exponents $n$ and $m$ while preserving the energy functional and steady states via fixed $q = m - n + 1$.
• Use of a timestepping scheme that refines resolution near singularities to detect finite-time touch-down events.
• Comparison of solution trajectories under perturbations to determine convergence to constant, periodic, or droplet steady states.

Experimental results

Research questions

• RQ1 Does the linear stability of periodic steady states predict their nonlinear stability or instability in numerical simulations?
• RQ2 Can heteroclinic orbits connecting different steady states (e.g., periodic to constant or droplet) be numerically observed and how do they depend on $n$ and $m$?
• RQ3 How do changes in mobility exponents $n$ and $m$ affect the occurrence and timing of finite-time touch-down singularities?
• RQ4 Do heteroclinic connections persist when mobility parameters are altered, even though the underlying energy landscape remains unchanged?
• RQ5 What is the role of energy minimization in determining the long-time behavior of perturbed solutions?

Key findings

• The constant steady state is asymptotically stable under perturbations, with solutions relaxing back to it over time.
• Perturbations of the positive periodic steady state evolve toward the constant steady state if perturbed in one direction, but lead to finite-time touch-down in the opposite direction.
• Heteroclinic orbits connecting periodic, constant, and droplet steady states are numerically observed, supporting the energy-based prediction of a mountain pass scenario.
• When mobility parameters $n$ and $m$ are varied while preserving $q = m - n + 1$, the shape of heteroclinic connections remains qualitatively similar, though timescales change significantly.
• The exponent $n$ influences whether touch-down occurs in finite or infinite time, and can determine the number of touch-down events per period (one or two).
• No evidence of spurious singularities was found, and adaptive timestepping confirmed that small timesteps were used only near genuine singularities, not due to numerical instability.