Related papers: High-speed kinks in a generalized discrete $\phi^4…
We show convergence of solutions of a convective Allen-Cahn equation for a given smooth and divergence free velocity field to a transport equation for an evolving interface in the case when the thickness of the diffuse interface tends to…
The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here we present new scalable MFS…
We leverage the proximal Galerkin algorithm (Keith and Surowiec, Foundations of Computational Mathematics, 2024, DOI: 10.1007/s10208-024-09681-8), a recently introduced mesh-independent algorithm, to obtain a high-order finite element…
In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a…
We solve a difficult problem involving velocity and acceleration components along oblique axes, and propose two problems of central force motion to be solved using oblique axes.
A dynamic crack tip equation of motion is proposed based on the autonomy of the near-tip nonlinear zone of scale $\ell_{nl}$, symmetry principles, causality and scaling arguments. Causality implies that the asymptotic linear-elastic fields…
A kinetic model for granular mixtures is considered to study three different non-equilibrium situations. The model is based on the equivalence between a gas of elastic hard spheres subjected to a drag force proportional to the particle…
A dynamic crack will travel in a straight path up to a material-dependent critical speed beyond which its path becomes erratic. Predicting this critical speed and discovering the origin of this instability are two outstanding problems in…
In this paper, we consider a non-linear fourth-order evolution equation of Cahn-Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order…
Physics-Informed Neural Networks (PINN) are a machine learning tool that can be used to solve direct and inverse problems related to models described by Partial Differential Equations. This paper proposes an adaptive inverse PINN applied to…
We give an exhaustive, non-perturbative classification of exact travelling-wave solutions of a perturbed sine-Gordon equation (on the real line or on the circle) which is used to describe the Josephson effect in the theory of…
A Lax pair consisting of the Forsyth-Hopf-Cole transformation and a linear differential (in time) - difference (in continuous space) equation generates the whole hierarchy of the Burgers equation and admits exact moving front solutions.…
We describe exact kink soliton solutions to nonlinear partial differential equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx} = A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality allows…
Recently we considered a stochastic discrete model which describes fronts of cells invading a wound \cite{KSS}. In the model cells can move, proliferate, and experience cell-cell adhesion. In this work we focus on a continuum description of…
This paper concerns classical nonlinear scalar field models on the real line. If the potential is a symmetric double-well, such a model admits static solutions called kinks and antikinks, which are perhaps the simplest examples of…
We consider a version of the classical Hamiltonian FPU (Fermi-Pasta-Ulam) problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition…
We investigate chimera synchronization of internal oscillator states in a ring of interacting particles, using the damped dc-driven Frenkel--Kontorova chain model as an example. In a system with a spatially periodic potential, a dc external…
We study the solutions of a friction oscillator subject to stiction. This discontinuous model is non-Filippov, and the concept of Filippov solution cannot be used. Furthermore some Carath\'eodory solutions are unphysical. Therefore we…
Nonlinear waves are a robust phenomenon observed in complex systems ranging from mechanics to ecology. Fronts are fundamental due to their robustness against perturbations and capacity to propagate one state over another. Controlling and…
It is shown that the kinematic equations governing steady motions of an ideal fibre-reinforced fluid in a curved stratum may be expressed entirely in terms of the intrinsic Gauss equation, which assumes the form of a partial differential…