Narrow Results

We consider, in a D-dimensional cylinder, a non-local evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider sub-critical temperatures, for which there are two local spatially homogeneous equilibria, and show a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibrium; i.e. the planar fronts: We show that an initial perturbation of a front that is sufficiently small in L² norm, and sufficiently localized yields a solution that relaxes to another front, selected by a conservation law, in the L¹ norm at an algebraic rate that we explicitly estimate. We also obtain rates for the relaxation in the L² norm and the rate of decrease of the excess free energy.

In the Robe’s restricted three-body problem, we have considered the motion of the test particle which is moving inside the outermost layer of the heterogeneous body. This heterogeneous body has N layers with different densities and is filled with viscous fluid. The test particle which is taken as the third (or infinitesimal) body is moving under the influence of the heterogeneous body (primary) and point mass (secondary). We are motivated from Ansari,¹ where the linear stability and other important properties of this specific model have been discussed. In this paper, we have extended their work and discussed here the nonlinear stability of the non-collinear stationary points $L_4$ and $L_5$. Therefore, using the Arnold–Moser theorem (Kolmogorov–Arnold–Moser theory), we have done our analysis on the nonlinear stability and obtained significant results.

Modeling networks of synaptically coupled neurons often leads to systems of integro-differential equations. Particularly interesting solutions in this context are traveling waves. We prove here that spectral stability of traveling waves implies their nonlinear stability in appropriate function spaces, and compare several recent Evans-function constructions that are useful tools when analyzing spectral stability.

We describe conditions under which higher-dimensional billiard models in bounded, convex regions are fully chaotic, generalizing the Bunimovich stadium to dimensions above two. An example is a three-dimensional stadium bounded by a cylinder and several planes; the combination of these elements may give rise to defocusing, allowing large chaotic regions in phase space. By studying families of marginally-stable periodic orbits that populate the residual part of phase space, we identify conditions under which a nonlinear instability mechanism arises in their vicinity. For particular geometries, this mechanism rather induces stable nonlinear oscillations, including in the form of whispering-gallery modes.

For the 𝔰𝔬(4) free rigid body the stability problem for the isolated equilibria has been completely solved using Lie-theoretical and topological arguments. For each case of nonlinear stability previously found, we construct a Lyapunov function. These Lyapunov functions are linear combinations of Mishchenko's constants of motion.

We study the existence and stability of periodic traveling-wave solutions for the quadratic and cubic nonlinear Schrödinger equations in one space dimension.

Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

We study the Bénard problem in the Boussinesq approximation with upper free surface in the presence of surface tension. Both buoyancy and Marangoni effect are taken into account. Defining a suitable energy functional, we obtain sufficient conditions for nonlinear exponential stability of the rest state.

We analyze a reduced 1D Vlasov–Maxwell system introduced recently in the physical literature for studying laser-plasma interaction. This system can be seen as a standard Vlasov equation in which the field is split into two terms: an electrostatic field obtained from Poisson's equation and a vector potential term satisfying a nonlinear wave equation. Both nonlinearities in the Poisson and wave equations are due to the coupling with the Vlasov equation through the charge density. We show global existence of weak solutions in the nonrelativistic case, and global existence of characteristic solutions in the quasi-relativistic case. Moreover, these solutions are uniquely characterized as fixed points of a certain operator. We also find a global energy functional for the system allowing us to obtain L^p-nonlinear stability of some particular equilibria in the periodic setting.

Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler^1,2 and was modeled by Keller and Segel.¹⁵ For a quasilinear hyperbolic–parabolic system that arises as a non-diffusive limit of the Keller–Segel model with nonlinear kinetics, we establish the existence and nonlinear stability of traveling wave solutions with large amplitudes. The numerical simulations are performed to show the stability of the traveling waves under various perturbations.

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed ϵ^-1 (ϵ > 0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter ϵ: for small ϵ the behavior is essentially the same as for the diffusive Fisher-KPP equation. However, for large ϵ the traveling front with minimal speed is discontinuous and travels at the maximal speed ϵ^-1. The traveling fronts with minimal speed are linearly stable in weighted L² spaces. We also prove local nonlinear stability of the traveling front with minimal speed when ϵ is smaller than the transition parameter.

Proceeding with a series of works (Refs. 12, 23–25) by the authors, this paper establishes the nonlinear asymptotic stability of traveling wave solutions of the Keller–Segel system with nonzero chemical diffusion and linear consumption rate, where the right asymptotic state of cell density is vacuum (zero) and the initial value is a perturbation with zero integral from the spatially shifted traveling wave. The main challenge of the problem is various singularities caused by the logarithmic sensitivity and the vacuum asymptotic state, which are overcome by a Hopf–Cole type transformation and the weighted energy estimates with an unbounded weight function introduced in the paper.

A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

This work presents a one-dimensional finite element formulation for nonlinear analysis of spaced framed structures with thin-walled cross-sections. Within the framework of updated Lagrangian formulation, the nonlinear displacement field of thin-walled cross-sections, which accounts for restrained warping as well as the second-order displacement terms due to large rotations, the equations of equilibrium are firstly derived for a straight beam element. Due to the nonlinear displacement field, the geometric potential of semitangential moment is obtained for both the torsion and bending moments. In such a way, the joint moment equilibrium conditions of adjacent non-collinear elements are ensured. Force recovering is performed according to the external stiffness approach. Material nonlinearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends and for this a corresponding plastic reduction matrix is determined. The interaction of element forces at the hinge and the possibility of elastic unloading are taken into account. The effectiveness of the numerical algorithm discussed is validated through the test problem.

This paper presents a one-dimensional (1D) finite element formulation for the nonlinear stability analysis of framed structures with semi-rigid (SR) connections. By applying the updated Lagrangian incremental formulation and the nonlinear displacement field of thin-walled cross sections, the equilibrium equations of a straight beam element are first developed. Force recovering is performed according to the external stiffness approach. Material nonlinearity is introduced for an elastic-perfectly plastic material through the plastic hinge formation at finite element ends. To account for the SR connection behavior, a special transformation procedure is developed. The effectiveness of the numerical algorithm discussed is validated through the test problems.

The geometrically nonlinear response of sandwich functionally graded cylindrical shells reinforced by orthogonal and/or spiral stiffeners and subjected to axial compressive loads is investigated in this paper. Two types of sandwich functionally graded material models are considered. The formulations are based on the Donnell shell theory considering geometrical nonlinearity and Pasternak’s elastic foundation. The improved Lekhnitskii’s smeared stiffener technique is used to account for the stiffener effects with both mechanical and thermal stresses. The results obtained indicate that the spiral stiffeners have significantly beneficial influences in comparison with orthogonal stiffeners on the nonlinear buckling behavior of shells. The relatively large effects of temperature change, geometrical and material parameters are also demonstrated in the numerical investigations.

Low-frequency hunting problems of high-speed railway vehicles frequently occur due to the complex operating environment and degradation of wheel–rail contact conditions, which significantly affect the running safety and ride comfort of high-speed trains (HSTs). This paper presents a numerical investigation of the influence of aerodynamic loads on the carbody low-frequency hunting behaviors of HST. Considering the effect of aerodynamic loads, a multi-body system dynamics model for a HST train is formulated and applied to reproduce the carbody low-frequency hunting behavior. The influence of aerodynamic loads and wheel–rail contact conditions on the nonlinear stability of HST is analyzed. The range of aerodynamic coefficients of different aerodynamic loads which can stimulate the low-frequency hunting behavior of HST is proposed. The results show that the aerodynamic loads have a prominent effect on the nonlinear stability of HSTs. The low-frequency hunting motion of the HST tail car can be motivated by the lift airflow generated during service operation with a high traveling speed. The running stability of HSTs is more easily influenced by the aerodynamic loads when wheels are reprofiled.

In the general case of the free rigid body, we will give a list of integrals of motion, which generate the set of Mishchenko's integrals. In the case of , we prove that there are 15 coordinate-type Cartan subalgebras which on a regular adjoint orbit give 15 Weyl group orbits of equilibria. These coordinate-type Cartan subalgebras are the analogs of the three axes of equilibria for the classical rigid body on . The nonlinear stability and instability of these equilibria is analyzed. In addition to these equilibria there are 10 other continuous families of equilibria.

We study the nonlinear stability of the equilibrium state associated with the initial boundary value problem for the Broadwell model with transonic boundary. Based on the Green's function of this linearized initial boundary value problem and on the understanding of its action on microscopic data, we establish the pointwise convergence of the solution toward the equilibrium state under small perturbation, via a solution representation and Picard-like iterations.