Non-symmetric Solutions Research Articles

Transitions between Localised Patterns with Different Spatial Symmetries in Non-Local Hyperbolic Models for Self-Organised Biological Aggregations

Pattern formation in biological aggregations is a topic of great interest, due to the complex spatial structure of various aggregations of cells/bacteria/animals that can be observed in nature. While many such aggregations look similar at the macroscopic level, they might differ in their microscopic spatial structure. However, the complexity of the non-linear and sometimes non-local interactions among individuals inside these aggregations makes it difficult to investigate these spatial structures. In this study, we investigate numerically the transitions between different spatial patterns of animal aggregations with various symmetries (even, odd or no symmetry) that characterise the microscopic distribution of individuals inside these aggregations. To this end, we construct a bifurcation diagram starting with perturbations of spatially homogeneous solutions with low, medium, and high amplitudes. For perturbations with low amplitudes, the bifurcating structures show transitions among even-symmetric, odd-symmetric, and non-symmetric solutions. For perturbations with large amplitudes, there are wide parameter regions with non-convergent solutions, characterised by oscillatory transitions between different relatively similar solutions. These numerical results emphasize: (i) the effect of nonlinear and non-local interactions on the microscopically different symmetric/non-symmetric structures of macroscopically similar ecological aggregations; (ii) the difficulty of developing continuation algorithms for this class of non-local models.

3D numerical simulation of an anisotropic bead type thermistor and multiplicity of solutions

We perform some 3D numerical experiments for the approximation of the solutions to a bead type thermistor problem. We consider the case of a diagonal anisotropic diffusion matrix whose jth entry is of the form |∂u/∂xj|pj−2∂u/∂xj, u being the temperature inside the thermistor and the exponents pj, 1≤j≤3, lie in the interval (1,+∞). We first show some existence results for different notions of solutions, prove a maximum principle for each type of solution, and study certain symmetry properties for these solutions in a bead type thermistor. These properties lead us to the introduction of a symmetric solution and we show the existence of such a solution.We have developed a numerical algorithm for the computation of the numerical solutions in a bead type thermistor. This algorithm combines a fixed-point technique with a standard finite element method (FEM). Some numerical tests have shown the existence of non-symmetric solutions and this leads to multiple many solutions (at least three). We discuss the numerical results obtained for different values of the exponents pj and the applied voltage on different meshes.

Existence of Non-symmetric Solutions to the Gaussian Minkowski Problem

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang et al. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge–Ampère type equation on the sphere.

Topology of the singularities of 3-RPR planar parallel robots

Given a general 3-RPR planar parallel robot with linear platforms, it is proven that there are no singularity-free paths between non-symmetrical direct kinematics solutions. We provide an alternative proof of this. We also provide a proof that shows that there is a singularity-free path between two symmetrical solutions if an appropriate condition is satisfied. This condition will be automatically satisfied for exactly one symmetrical pair of solutions if there are four real solutions to the direct kinematics. The topology of the kinematic singularities of these robots is described. We prove that the complement of the singularity space in the domain of the kinematic map for such robots consists of three connected components. An analysis of special robots is also provided, i.e. when the anchor points of the moving platform and the fixed platform have the same cross-ratios. For this case we show that, the singularity-free space consists of four connected components, no direct kinematics solutions can be connected without crossing the singularity surface and that the absolute value of the determinant of the Jacobian of the kinematic map evaluates to the same number for any of the direct kinematics solutions. The topological analysis of the singularities of 3-RPR planar parallel robots with linear platforms is based on the fibers of the natural map SE(2)→S1 restricted to the singularity set of the kinematic map, and this is a conic fibration. In fact this is also the case when one or both platforms are triangular. We use similar methodology as in the linear platforms case and show that in the case that one or both platforms are triangles the singularity-free space has two connected components. This result was already proven when both platforms are triangles but here a different approach is followed.

Flame propagation in narrow horizontal channels: Impact of the gravity field on the flame shape

In this paper, we study the effect of gravity, or buoyancy forces, on the structure of flames propagating in horizontal channels. It is shown that there are two mechanisms for the appearance of non-symmetric flames. The first, more obvious, is related to buoyancy, when a hotter gas of lower density tends to be located in the upper half of the channel. However, there is a second mechanism associated with the intrinsic flame instabilities, which also can cause the loss of flame symmetry. This mechanism can, at certain values of the parameters, act in the opposite direction, when hotter gases are enclosed in the lower part of the channel. In this case, two stable non-symmetric solutions may exist in the presence of gravitational forces and the establishment of one or another configuration depends on the initial conditions. The stability of these solutions is demonstrated by time-dependent computations.

Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein–Gordon Equation

This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.

Non-symmetric blowing-up solutions for a class of Liouville equations in the ball

We are concerned with the existence of blowing-up solutions to the following boundary value problem: −Δv = λV(x)ev in B1, v = 0 on ∂B1, where B1 is the unit ball in R2, V(x) is a positive smooth potential, and λ > 0 is a small parameter. We assume that the potential V satisfies some suitable assumptions in terms of the second and the fourth derivatives at 0, and we find a solution that exhibits a non-symmetric blow-up profile as λ → 0+.

Mixed Convection Heat Transfer and Entropy Generation in a Water-filled Square Cavity Partially Heated from Below: The Effects of Richardson and Prandtl Numbers

In the present study, fluid flow, heat transfer, and entropy generation for mixed convection inside a water-filled square cavity were investigated numerically. The sidewalls of the cavity, which move upwards, are kept at low-temperature Tc while only a part in the center of the bottom wall is kept at high-temperature Th and the remaining parts are kept adiabatic. The governing equations, in stream function–vorticity form, are discretized and solved using the finite difference method. Particular attention was paid to the influence of the Prandtl numbers of 5.534, 3.045 and 2, corresponding respectively to the water temperatures of 303.15 K, 333.15 K and 363.15 K. The numerical results are presented in the form of streamlines, isotherms, and entropy generation contours for different values of the Richardson numbers at an arbitrary Reynolds number Re=102. Besides this, the evolution of the average Nusselt number and the average entropy generation is also reported. The obtained results show interesting behaviors of the flow and thermal fields, which mainly involve stable symmetric and non-symmetric steady-state solutions, as well as unsteady regimes, depending on specific values of the Richardson and Prandtl numbers. It is additionally observed that the average Nusselt number increases and the average entropy generation decreases when both the Richardson and Prandtl numbers increase.

Uniqueness and Bifurcation Branches for Planar Steady Navier\u2013Stokes Equations Under Navier Boundary Conditions

The stationary Navier–Stokes equations under Navier boundary conditions are considered in a square. The uniqueness of solutions is studied in dependence of the Reynolds number and of the strength of the external force. For some particular forcing, it is shown that uniqueness persists on some continuous branch of solutions, when these quantities become arbitrarily large. On the other hand, for a different forcing, a branch of symmetric solutions is shown to bifurcate, giving rise to a secondary branch of nonsymmetric solutions. This proof is computer-assisted, based on a local representation of branches as analytic arcs.

Critical conditions for non-symmetric flame propagation in narrow channels: Influence of the flow rate, the thermal expansion, the Lewis number and heat-losses

This paper presents a numerical study of the propagation of isobaric premixed flames in narrow channels, focusing primarily on the symmetry of solutions. Because of the complexity of the problem, only few but very important effects such as the thermal expansion, the Lewis number and heat losses are included in the formulation and only a planar channel geometry is investigated. It is demonstrated that symmetric and non-symmetric steady state flames can exist simultaneously, for certain ranges of the parameters. The parametric dependence of the corresponding bifurcation points is also investigated. It is found that both subcritical and supercritical bifurcations are possible for different values of the parameters. The steady-state results are successfully compared with direct time-dependent simulations. These simulations show that, as a general rule, in cases where both types of solutions exist simultaneously, the non-symmetric solutions are stable while the symmetric solutions are unstable.

Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity. III: nonspherical Schwarzschild waves and singularities at null infinity

We extend earlier numerical and analytical considerations of the conformally invariant wave equation on a Schwarzschild background from the case of spherically symmetric solutions, discussed in Frauendiener and Hennig (2017 Class. Quantum Grav. 34 045005), to the case of general, nonsymmetric solutions. A key element of our approach is the modern standard representation of spacelike infinity as a cylinder. With a decomposition into spherical harmonics, we reduce the four-dimensional wave equation to a family of two-dimensional equations. These equations can be used to study the behaviour at the cylinder, where the solutions turn out to have, in general, logarithmic singularities at infinitely many orders. We derive regularity conditions that may be imposed on the initial data, in order to avoid the first singular terms. We then demonstrate that the fully pseudospectral time evolution scheme can be applied to this problem leading to a highly accurate numerical reconstruction of the nonsymmetric solutions. We are particularly interested in the behaviour of the solutions at future null infinity, and we numerically show that the singularities spread to null infinity from the critical set, where the cylinder approaches null infinity. The observed numerical behaviour is consistent with similar logarithmic singularities found analytically on the critical set. Finally, we demonstrate that even solutions with singularities at low orders can be obtained with high accuracy by virtue of a coordinate transformation that converts solutions with logarithmic singularities into smooth solutions.

Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles.

We present numerical solutions to the Landau-de Gennes free-energy model under the one-constant approximation for systems of single and double spherical colloidal particles immersed in an otherwise uniformly aligned nematic liquid crystal. A perfect homeotropic surface anchoring of liquid-crystal molecules on the spherical surface is considered. A large parameter space is carefully examined, including those in the free-energy model and those describing the dimer configurations and the background liquid-crystal orientation. The stability of the resulting liquid-crystal defects appearing in the neighborhood of the colloidal dimer pair is analyzed in light of the numerical results for their free energies. A number of scenarios are considered: a free dimer pair in a nematic fluid where the free-energy ground states are described in terms of a phase diagram, and a constrained dimer pair where the interparticle distance and the relative orientation of the distance vector to the nematic director can be manipulated. We pay particular attention to the nonsymmetric solutions, which yield several metastable defect states that can be observed in real systems. The high-precision numerical calculations are based on a spectral method, which is an enabling factor that allows us to compare the subtle difference in the free energies of different defect structures.

Non-symmetrical separated flow along the parabolic wing

Inviscid incompressible separated flow along the parabolic wing is explored. The partition is placed in the symmetry plane. The problem is considered without an assumption of flow symmetry. Self-similar solution is constructed. Both symmetrical and non-symmetrical solutions are observed. Fields of self-similar trajectories and stability are analysed.

Symmetrical or Non-Symmetrical Debonds at Fiber–Matrix Interfaces: A Study by BEM and Finite Fracture Mechanics on Elastic Interfaces

A recently proposed criterion is used to study the behavior of debonds produced at a fiber–matrix interface. The criterion is based on the Linear Elastic–(Perfectly) Brittle Interface Model (LEBIM) combined with a Finite Fracture Mechanics (FFM) approach, where the stress and energy criteria are suitably coupled. Special attention is given to the discussion about the symmetry of the debond onset and growth in an isolated single fiber specimen under uniaxial transverse tension. A common composite material system, glass fiber–epoxy matrix, is considered. The present methodology uses a two-dimensional (2D) Boundary Element Method (BEM) code to carry out the analysis of interface failure. The present results show that a non-symmetrical interface crack configuration (debonds at one side only) is produced by a lower critical remote load than the symmetrical case (debonds at both sides). Thus, the non-symmetrical solution is the preferred one, which agrees with the experimental evidences found in the literature.

Stability analysis solutions of the nonlinear modified Degasperis–Procesi water wave equation

In the present study, the solitary wave solutions of modified Degasperis–Procesi equation are developed. Unlike the standard Degasperis–Procesi equation, where multi-peakon solutions arise, the modification caused a change in the characteristic of these peakon solutions and changed it to bell-shaped solitons. By using the extended auxiliary equation method, we deduced some new soliton solutions of the fourth-order nonlinear modified Degasperis–Procesi equation with constant coefficient. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, weiestrass elliptic function solutions and triangular function solutions. We discuss the stability analysis for these solutions.

Effects of stoichiometry on premixed flames propagating in narrow channels: symmetry-breaking bifurcations

Recent studies within the diffusive–thermal (constant-density) approximation have shown that, for premixed flames freely propagating in narrow adiabatic channels, the instabilities induced by differential diffusion may result in non-symmetric solutions and/or oscillating and rotating propagation modes. This has been shown in the context of lean mixtures, for which a single species transport equation with a single Lewis number (corresponding to the ratio of thermal to molecular diffusivity of the deficient reactant) can be used to describe the flame propagation problem. In the present work we extend the analysis to mixtures of any equivalence ratio. To this end, we consider a two-reactant model, where the different diffusivities of the two reactants introduce two different Lewis numbers. Steady-state computations and linear stability analysis are carried out for mixtures with large disparity between the Lewis number of the fuel () and the oxidiser (), such as hydrogen–oxygen systems. It is shown that both differential diffusion and preferential diffusion have influence on the stability of the symmetric flame shape. For sufficiently lean and rich mixtures, the flame behaves as dictated by the Lewis number of the deficient reactant, i.e. the flame destabilises toward non-symmetric solutions for large mass flow rates when the mentioned Lewis number is less than one. In near-stoichiometric mixtures the stability of the symmetric flame depends on a weighted average value of and . In particular, the symmetric solution is stable for large mass flow rates because of the difficulty found by the less diffusive reactant to reach the reactive zone of the flame.

Stability analysis of solutions for the sixth-order nonlinear Boussinesq water wave equations in two-dimensions and its applications

In the present study, we are concerned with the generalized Boussinesq equation including the singular sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional sixth-order nonlinear Boussinesq equation with constant coefficients. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions is discussed.

Self-Similarity in a Thin Film Muskat Problem

The large time behavior of non-negative weak solutions to a thin film approximation of the two-phase Muskat problem is studied. A classification of self-similar solutions is first provided: there is always a unique even self-similar solution while a continuum of non-symmetric self-similar solutions exist for certain fluid configurations. Despite this non-uniqueness, convergence of all non-negative weak solutions towards a self-similar solution is proved.

Investigation of symmetry breaking in periodic gravity–capillary waves

In this paper, fully nonlinear non-symmetric periodic gravity–capillary waves propagating at the surface of an inviscid and incompressible fluid are investigated. This problem was pioneered analytically by Zufiria (J. Fluid Mech., vol. 184, 1987c, pp. 183–206) and numerically by Shimizu & Shōji (Japan J. Ind. Appl. Maths, vol. 29 (2), 2012, pp. 331–353). We use a numerical method based on conformal mapping and series truncation to search for new solutions other than those shown in Zufiria (1987c) and Shimizu & Shōji (2012). It is found that, in the case of infinite-depth, non-symmetric waves with two to seven peaks within one wavelength exist and they all appear via symmetry-breaking bifurcations. Fully exploring these waves by changing the parameters yields the discovery of new types of non-symmetric solutions which form isolated branches without symmetry-breaking points. The existence of non-symmetric waves in water of finite depth is also confirmed, by using the value of the streamfunction at the bottom as the continuation parameter.

Basin of Attraction of Solutions with Pattern Formation in Slow-Fast Reaction-Diffusion Systems.

This article is devoted to the characterization of the basin of attraction of pattern solutions for some slow-fast reaction-diffusion systems with a symmetric property and an underlying oscillatory reaction part. We characterize some subsets of initial conditions that prevent the dynamical system to evolve asymptotically toward solutions which are homogeneous in space. We also perform numerical simulations that illustrate theoretical results and give rise to symmetric and non-symmetric pattern solutions. We obtain these last solutions by choosing particular random initial conditions.