Stability of Gaussian-type soliton in the cubic–quintic nonlinear media with fourth-order diffraction and $$mathcal {PT}$$ PT -symmetric potentials

Nonlinear Dynamics - We report on the existence and stability of Gaussian-type soliton in the nonlinear Schrödinger (NLS) equation with interplay of cubic–quintic nonlinearity,...     

Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under $${{varvec{mathcal {P}}}}{varvec{mathcal {T}}}$$ P T -symmetric potentials

Nonlinear Dynamics - Gaussian spatial soliton solutions of both the constant-coefficient and variable-coefficient (2 + 1)-dimensional nonlinear Schrödinger equations in...     

Broken and unbroken $$varvec{mathcal {PT}}$$ PT -symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation

Nonlinear Dynamics - In this letter, we obtain multi-soliton solutions in terms of ratio of ordinary determinants for semi-discrete nonlocal nonlinear Schrödinger (sd-NNLS) equation by...     

The stability of two-dimensional spatial solitons in cubic–quintic–septimal nonlinear media with different diffractions and $${\varvec{\mathcal {}}}$$-symmetric potentials

A (2+1)-dimensional nonlinear Schrödinger equation in cubic–quintic–septimal nonlinear media with different diffractions and \({\mathcal {PT}}\)-symmetric potentials is studied, and (2+1)-dimensional spatial solitons are derived. The stable region of analytical spatial solitons is discussed by means of the eigenvalue method. The direct numerical simulation indicates that analytical spatial soliton solutions stably evolve within stable region in the media of focusing septimal and focusing or defocusing cubic nonlinearities with disappearing quintic nonlinearity under the 2D extended Scarf II potential. However, under the extended \({\mathcal {PT}}\)-symmetric potential with \(p=2\) and \(p=3\), analytical spatial soliton solutions stably evolve within stable region in the media of focusing quintic and septimal nonlinearities with defocusing cubic nonlinearity. In other cases, analytical spatial soliton solutions cannot sustain their original shapes, and they are distorted and broken up and finally decay into noise.     

Computational analysis on Hopf bifurcation and stability for a consumer–resource model with nonlinear functional response

We consider a consumer–resource model with nonlinear functional response and reaction–diffusion terms. By taking the growth rate of the resource as the parameter, we give a computational and theoretical analysis on Hopf bifurcation emitting from the positive equilibrium for the model and discuss the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions by space decomposition and vector operation techniques. It is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Some numerical examples are presented to support and illustrate our theoretical analysis.     

Bifurcation and stability analysis in predator–prey model with a stage-structure for predator

A predator–prey system with Holling type II functional response and stage-structure for predator is presented. The stability and Hopf bifurcation of this model are studied by analyzing the associated characteristic transcendental equation. Further, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from positive equilibrium is derived by the normal form theory and center manifold argument. Some numerical simulations are also given to illustrate our results.     

Dynamics and bifurcations in the weak electrolyte model for electroconvection of nematic liquid crystals: a Ginzburg–Landau approach

In this paper we present the results of a bifurcation study of the weak electrolyte model for nematic electroconvection, for values of the parameters including experimentally measured values of the nematic I52. The linear stability analysis shows the existence of primary bifurcations of Hopf type, involving normal as well as oblique rolls. The weakly nonlinear analysis is performed using four globally coupled complex Ginzburg–Landau equations for the waves' envelopes. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with $O(2)\times O(2)$ symmetry. A rich variety of stable waves, as well as more complex spatiotemporal dynamics is predicted at onset. A temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, is identified. Eckhaus stability boundaries for travelling waves are also determined. The methods developed in this paper provide a systematic investigation of nonlinear physical mechanisms generating the patterns observed experimentally, and can be generalized to any two-dimensional anisotropic systems with translational and reflectional symmetry.     

$$\pmb {H_\infty }$$ H ∞ fusion estimation of time-delayed nonlinear systems with energy constraints: the finite-horizon case

Nonlinear Dynamics - The fusion estimation issue of sensor networks is investigated for nonlinear time-varying systems with energy constraints, time delays as well as packet loss. For the addressed...     

Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on $${\mathbb{R}^3}$$

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.     

Dynamic stability of shallow arch with elastic supports—application in the dynamic stability analysis of inner winding of transformer during short circuit

In this paper, the dynamic stability of a shallow arch with elastic supports subjected to impulsive load is used as a theoretical model to investigate the dynamic stability problem of inner windings of power transformer under short-circuit condition. Firstly, the series solution representing the equilibrium configurations of a shallow arch is obtained by solving the corresponding non-linear integration-differential equation. The local stability of each equilibrium configuration is discussed, and the sufficient condition for stability of the shallow arch system as well as the critical load against snap-through is obtained. Secondly, the equivalent relation between short-circuit load and impulsive one, and the electrical forces transferred pattern between the coils of inner windings are assumed. Then the results of the shallow arch model are applied to the case of the inner winding of transformer and the formulas for computing critical electromagnetic force and the dynamic stability criterion of the inner windings are established. Finally, examples are offered and the theoretical results are shown to agree well with the experimental ones.     

The use of the “trident” model in the analysis of plastic zones near crack tips and corner points

The paper deals with calculation of a plastic zone near a crack tip in a homogeneous elastoplastic solid and near a corner point of the boundary of this solid. The calculations are conducted for a solid subject to plane strain and within the framework of models with plastic strips. It is shown that in comparison with the widely used model with two straight slip-lines, the process of plastic deformation is described by the “trident” model more accurately. The results of calculations of the plastic zone by the “trident” model that correspond to different stages of the development of plastic deformation are given for a crack of normal separation in a quasibrittle material. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 95–100, March, 2000.     

Evolution behaviour of kink breathers and lump- $$\pmb {M}$$ M -solitons ( $$\pmb {M\rightarrow \infty }$$ M → ∞ ) for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Nonlinear Dynamics - In this paper, some novel lump solutions and interaction phenomenon between lump and kink M-soliton are investigated. Firstly, we study the evolution and degeneration behaviour...     

Solution and asymptotic analysis of a boundary value problem in the spring–mass model of running

We consider the classic spring–mass model of running which is built upon an inverted elastic pendulum. In a natural way, there arises an interesting boundary value problem for the governing system of two nonlinear ordinary differential equations. It requires us to choose the stiffness to ascertain that after a complete step, the spring returns to its equilibrium position. Motivated by numerical calculations and real data, we conduct a rigorous asymptotic analysis in terms of the Poicaré–Lindstedt series. The perturbation expansion is furnished by an interplay of two time scales what has an significant impact on the order of convergence. Further, we use these asymptotic estimates to prove that there exists a unique solution to the aforementioned boundary value problem and provide an approximation to the sought stiffness. Our results rigorously explain several observations made by other researchers concerning the dependence of stiffness on the initial angle of the stride and its velocity. The theory is illustrated with a number of numerical calculations.

     

Painlevé analysis,auto-Bäcklund transformation and analytic solutions of a (2 $$+$$ + 1)-dimensional generalized Burgers system with the variable coefficients in a fluid

Nonlinear Dynamics - Burgers-type equations are used to describe certain phenomena in gas dynamics, traffic flow, plasma astrophysics and ocean dynamics. In this paper, a (2 $$+$$ 1)-dimensional...     

The $$\varvec{N}$$ N -soliton,fusion, rational and breather solutions of two extensions of the (2+1)-dimensional Bogoyavlenskii–Schieff equation

Nonlinear Dynamics - The aim of this work is to analyze and explore the dynamics of two extensions of the Bogoyavlenskii–Schieff equation. The Hirota bilinear method is applied to the...