Under certain conditions the motion of superconducting vortices is primarily governed by an instability. We consider an averaged model, for this phenomenon, describing the motion of large numbers of such vortices. The model equations are parabolic, and, in one spatial dimension $x$, take the form \begin{eqnarray*} {H_2}_t &=& \frac{\partial}{\partial x} ( | {H_3} {H_2}_x-{H_2} {H_3}_x | {H_2}_x ), \\ {H_3}_t &=& \frac{\partial}{\partial x} ( | {H_3} {H_2}_x-{H_2} {H_3}_x | {H_3}_x ). \end{eqnarray*} where $H_2$ and $H_3$ are components of the magnetic field in the $y$ and $z$ directions respectively. These equations have an extremely rich group of symmetries and a correspondingly large class of similarity reductions. In this work, we look for non-trivial steady solutions to the model, deduce their stability and use a numerical method to calculate time-dependent solutions. We then apply Lie Group based similarity methods to calculate a complete catalogue of the model's similarity reductions and use this to investigate a number of its physically important similarity solutions. These describe the short time response of the superconductor as a current or magnetic field is switched on (or off).

In the limit of small activator diffusivity, the stability of a one-spike solution to the shadow Gierer–Meinhardt activator-inhibitor system is studied for various ranges of the reaction-time constant $\tau$ associated with the inhibitor field dynamics. By analyzing the spectrum of the eigenvalue problem associated with the linearization around a one-spike solution, it is proved, for a certain parameter regime, that a complex conjugate pair of eigenvalues crosses into the unstable right half-plane $Re(\lambda) > 0$ as $\tau$ increases past a critical value $\tau_0$. For this parameter regime, it is proved that there are exactly two eigenvalues in the right half-plane when $\tau > \tau_0$ and none when $0 \leq \tau < \tau_0$. It is shown numerically that this critical value of $\tau$ represents the onset of an oscillatory instability in the height of the spike. For other parameter regimes, a similar Hopf bifurcation is confirmed numerically. Full numerical solutions to the shadow problem are computed for a spike that is initially centred at the origin of a radially symmetric domain. Different types of large-scale oscillatory motions for the height of a spike are observed numerically for values of $\tau$ well beyond $\tau_0$.

Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.

We describe various blow-up patterns for the fourth-order one-dimensional semilinear parabolic equation \[ u_t = - u_{xxxx} + \b [(u_x)^3]_x + e^{u} \] with a parameter $\b \ge 0$, which is a model equation from explosion-convection theory. Unlike the classical Frank-Kamenetskii equation $u_t=u_{xx} +e^u$ (a solid fuel model), by using analytical and numerical evidence, we show that the generic blow-up in this fourth-order problem is described by a similarity solution $u_*(x,t) = -\ln(T\,{-}\,t) \,{+}\, f_1(x/(T\,{-}\,t)^{1/4})(T>0$ is the blow-up time), with a non-trivial profile $f_1 \not \equiv 0$. Numerical solution of the PDE shows convergence to the self-similar solution with the profile $f_1$ from a wide variety of initial data. We also construct a countable subset of other, not self-similar, blow-up patterns by using a spectral analysis of an associated linearized operator and matching with similarity solutions of a first-order Hamilton–Jacobi equation.