Nonlinear Sciences

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter a which controls the strength of advection. For solutions on the infinite domain we find a new critical value a c =0.6890665337007457… below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit into the line that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We find a new exact analytical collapsing solution at a=1/2 as well as prove the existence of a leading order complex singularity for general values of a in the analytical continuation of the solution from the real spatial coordinate into the complex plane. This singularity controls the leading order behaviour of the collapsing solution. For a c <a≤1 , we find a blow-up solution in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For a≳1.3 , we find that the solution exists globally with exponential-like growth of the solution amplitude in time. We also consider the case of periodic boundary conditions. We identify collapsing solutions for a< a c which are similar to the real line case. For a c <a≤0.95 , we find new blow-up solutions which are neither expanding nor collapsing. For a≥1, we identify a global existence of solutions.

The two major effects observed in collisions of the continuum ϕ 4 kinks are (i) the existence of critical collision velocity above which the kinks always emerge from the collision and (ii) the existence of the escape windows for multi-bounce collisions with the velocity below the critical one, associated with the energy exchange between the kink's internal and translational modes. The potential merger (for sufficiently low collision speeds) of the kink and antikink produces a bion with oscillation frequency ω B , which constantly radiates energy, since its higher harmonics are always within the phonon spectrum. Similar effects have been observed in the discrete ϕ 4 kink-antikink collisions for relatively weak discreteness. Here we analyze kinks colliding with their mirror image antikinks in the regime of strong discreteness considering an exceptional discretization of the ϕ 4 field equation where the static Peierls-Nabarro potential is precisely zero and the not-too-fast kinks can propagate practically radiating no energy. Several new effects are observed in this case, originating from the fact that the phonon band width is small for strongly discrete lattices and for even higher discreteness an inversion of the phonon spectrum takes place with the short waves becoming low-frequency waves. When the phonon band is narrow, not a bion but a discrete breather with frequency ω DB and all higher harmonics outside the phonon band is formed. When the phonon spectrum is inverted, the kink and antikink become mutually repulsive solitary waves with oscillatory tails, and their collision is possible only for velocities above a threshold value sufficient to overcome their repulsion.

We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations in the presence of the cubic loss and in the presence of the quintic loss.

A thring is a recent addition to the zoo of spiral wave phenomena found in excitable media and consists of a scroll ring that is threaded by a pair of counter-rotating scroll waves. This arrangement behaves like a particle that swims through the medium. Here, we present the first results on the dynamics, interaction and collective behaviour of several thrings via numerical simulation of the reaction-diffusion equations that model thrings created in chemical experiments. We reveal an attraction between two thrings that leads to a stable bound pair that thwarts their individual locomotion. Furthermore, such a pair emits waves at a higher frequency than a single thring, which protects the pair from the advances of any other thring and rules out the formation of a triplet bound state. As a result, the long-term evolution of a colony of thrings ultimately yields an unusual frozen nonequilibrium state consisting of a collection of pairs accompanied by isolated thrings that are inhibited from further motion by the waves emanating from the pairs.

We examine travelling wave solutions of the reaction-diffusion equation, ∂ t u=R(u)+ ∂ x [D(u) ∂ x u] , with a Stefan-like condition at the edge of the moving front. With only a few assumptions on R(u) and D(u) , a variety of new compactly supported travelling waves arise in this Reaction-Diffusion Stefan model. While other reaction-diffusion models admit compactly supported travelling waves for a unique wavespeed, we show that compactly supported travelling waves in the Reaction-Diffusion Stefan model exist over a range of wavespeeds. Furthermore, we determine the necessary conditions on R(u) and D(u) for which compactly supported travelling waves exist for all wavespeeds. Using asymptotic analysis in various distinguished limits of the wavespeed, we obtain approximate solutions of these travelling waves, agreeing with numerical simulations with high accuracy.

We present the Complex Envelope Variable Approximation (CEVA) as the very useful and compact method for the analysis of the essentially nonlinear dynamical systems. It allows us to study both the stationary and non-stationary dynamics even in the cases, when any small parameters are absent in the initial problem. It is notable that the CEVA admits the analysis of the nonlinear normal modes and their resonant interactions in the discrete systems without any restrictions on the oscillation amplitudes. In this paper we formulate the CEVA's formalism and demonstrate some non-trivial examples of its application. The advantages of the method and possible problems are briefly discussed.

In this paper, we introduce a three-component Schnakenberg model. Its key feature is that it has a solution consisting of N spikes that undergoes a Hopf bifurcation with respect to N distinct modes nearly simultaneously. This results in complex oscillatory dynamics of the spikes, not seen in typical two-component models. For parameter values above the Hopf bifurcations, we derive reduced equations of motion which consist of coupled ordinary differential equations (ODEs) of order 2N for spike positions and their velocities. These ODEs fully describe the slow-time evolution of the spikes near the Hopf bifurcations. We then apply the method of multiple scales to the resulting ODEs to derive long-time dynamics. For a single spike, we find that its long-time motion consists of oscillations near the steady-state whose amplitude can be computed explicitly. For two spikes, the long-time behaviour can be either in-phase or out-of-phase oscillations. Both in and out of phase orbits are stable, coexist for the same parameter values, and the fate of orbit depends solely on the initial conditions. Further away from the Hopf bifurcation point, we offer numerical experiments indicating the existence of highly complex and chaotic oscillations.

Compressed cylindrical shells are common in our daily life, such as the diamond shape in rolled-up sleeves, crumpled aluminum cans, and retreated package of now defunct drinking straws. The kind of deformation is formally called the Yoshimura pattern. However, there are many other equally prevalent modes of deformation, depending on the relative size of radius between the shell and its inner core, the thickness and rigidity and plasticity of the shell, etc. To elucidate the phase diagram for these modes, we combine molecular dynamics simulations and experiments to study the energetic, mechanical, and morphological responses of a compressed cylindrical shell with a hard core.

Configurational entropy (CE) consists of a family of entropic measures of information used to describe the shape complexity of spatially-localized functions with respect to a set of parameters. We obtain the Differential Configurational Entropy (DCE) for similariton waves traveling in tapered graded-index optical waveguides modeled by a generalized nonlinear Schrödinger equation. It is found that for similariton's widths lying within a certain range, DCE attains minimum saturation values as the nonlinear wave evolves along the effective propagation variable ζ(t) . In particular, saturation is achieved earlier for lower values of the width, which we show correspond to global minima of the DCE. Such low entropic values lead to minimum dispersion of momentum modes as the similariton waves propagate along tapered graded-index waveguides, and should be of importance in guiding their design.

Turing patterns in self-focussing nonlinear optical cavities pumped by beams carrying orbital angular momentum (OAM) m are shown to rotate with an angular velocity ω=2m/ R 2 on rings of radii R . We verify this prediction in 1D models on a ring and for 2D Laguerre-Gaussian and top-hat pumps with OAM. Full control over the angular velocity of the pattern in the range −2m/ R 2 ≤ω≤2m/ R 2 is obtained by using cylindrical vector beam pumps that consist of orthogonally polarized eigenmodes with equal and opposite OAM. Using Poincaré beams that consist of orthogonally polarized eigenmodes with different magnitudes of OAM, the resultant angular velocity is ω=( m L + m R )/ R 2 , where m L , m R are the OAMs of the eigenmodes, assuming good overlap between the eigenmodes. If there is no, or very little, overlap between the modes then concentric Turing pattern rings, each with angular velocity ω=2 m L,R / R 2 will result. This can lead to, for example, concentric, counter-rotating Turing patterns creating an 'optical peppermill'-type structure. Full control over the speeds of multiple rings has potential applications in particle manipulation and stretching, atom trapping, and circular transport of cold atoms and BEC wavepackets.