arXiv subjects

Christopher J. Lustri

Publications and source records attributed to Christopher J. Lustri.

23 records · Page 2Linked to original sources

arXiv2018

Generalized Solitary Waves in a Finite-Difference Korteweg-de Vries Equation

Generalized solitary waves with exponentially small non-decaying far field oscillations have been studied in a range of singularly-perturbed differential equations, including higher-order Korteweg-de Vries (KdV) equations. Many of these studies used exponential asymptotics to compute the behaviour of the oscillations, revealing that they appear in the solution as special curves known as Stokes lines are crossed. Recent studies have identified similar behaviour in solutions to difference equations. Motivated by these studies, the seventh-order KdV and a hierarchy of higher-order KdV equations are investigated, identifying conditions which produce generalized solitary wave solutions. These results form a foundation for the study of infinite-order differential equations, which are used as a model for studying lattice equations. Finally, a lattice KdV equation is generated using finite-difference discretization, in which a lattice generalized solitary wave solution is found.

math-ph↗

arXiv2018

Selection of a Hele-Shaw bubble via exponential asymptotics

The well-studied selection problems involving Saffman-Taylor fingers or Taylor-Saffman bubbles in a Hele-Shaw channel are prototype examples of pattern selection. Exact solutions to the corresponding zero-surface-tension problems exist for an arbitrary finger or bubble speed, but the addition of surface tension leads to a discrete set of solution branches, all of which approach a single solution in the limit the surface tension vanishes. In this sense, the surface tension selects a single physically meaningful solution from the continuum of zero-surface-tension solutions. Recently, we provided numerical evidence to suggest the selection problem for a bubble propagating in an unbounded Hele-Shaw cell behaves in an analogous way to other finger and bubble problems in a Hele-Shaw channel; however, the selection of the ratio of bubble speeds to background velocity appears to follow a very different surface tension scaling to the channel cases. Here we apply techniques in exponential asymptotics to solve the selection problem analytically, confirming the numerical results, including the predicted surface tension scaling laws. Further, our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. These results have ramifications for exotic-shaped Saffman-Taylor fingers as well as for the time-dependent evolution of bubbles propagating in Hele-Shaw cells.

physics.flu-dyn↗

arXiv2018

Nanoptera in a Period-2 Toda Chain

We study asymptotic solutions to a singularly-perturbed, period-2 Toda lattice and use exponential asymptotics to examine `nanoptera', which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains. With this approach, we isolate the exponentially small, constant-amplitude waves, and we elucidate the dynamics of these waves in terms of the Stokes phenomenon. We find a simple asymptotic expression for the waves, and we study configurations in which these waves vanish, producing localized solitary-wave solutions. In the limit of small mass ratio, we derive a simple anti-resonance condition for the manifestation these wave-free solutions.

nlin.PS↗

arXiv2018

Three-dimensional capillary waves due to a submerged source with small surface tension

Steady and unsteady linearised flow past a submerged source are studied in the small-surface-tension limit, in the absence of gravitational effects. The free-surface capillary waves generated are exponentially small in the surface tension, and are determined using the theory of exponential asymptotics. In the steady problem, capillary waves are found to extend upstream from the source, switching on across curves on the free surface known as Stokes lines. Asymptotic predictions and compared with computational solutions for the position of the free surface. In the unsteady problem, transient effects cause the solution to display more complicated asymptotic behaviour, such as higher-order Stokes lines. The theory of exponential asymptotics is applied to show how the capillary waves evolve over time, and eventually tend to the steady solution.

physics.flu-dyn↗

arXiv2016

Multiple scales and matched asymptotic expansions for the discrete logistic equation

In this paper, we combine the method of multiple scales and the method of matched asymptotic expansions to construct uniformly-valid asymptotic solutions to autonomous and non-autonomous difference equations in the neighbourhood of a period-doubling bifurcation. In each case, we begin by constructing multiple scales approximations in which the slow time scale is treated as a continuum variable, leading to difference-differential equations. The resultant approximations fail to be asymptotic at late time, due to behaviour on the slow time scale, it is necessary to eliminate the effects of the fast time scale in order to find the late time rescaling, but there are then no difficulties with applying the method of matched asymptotic expansions. The methods that we develop lead to a general strategy for obtaining asymptotic solutions to singularly-perturbed difference equations, and we discuss clear indicators of when multiple scales, matched asymptotic expansions, or a combined approach might be appropriate.

math.DS↗