共查询到20条相似文献,搜索用时 31 毫秒
1.
Jonathan A. Sherratt 《Advances in Computational Mathematics》2013,39(1):175-192
A variety of numerical methods are available for determining the stability of a given solution of a partial differential equation. However for a family of solutions, calculation of boundaries in parameter space between stable and unstable solutions remains a major challenge. This paper describes an algorithm for the calculation of such stability boundaries, for the case of periodic travelling wave solutions of spatially extended local dynamical systems. The algorithm is based on numerical continuation of the spectrum. It is implemented in a fully automated way by the software package wavetrain, and two examples of its use are presented. One example is the Klausmeier model for banded vegetation in semi-arid environments, for which the change in stability is of Eckhaus (sideband) type; the other is the two-component Oregonator model for the photosensitive Belousov–Zhabotinskii reaction, for which the change in stability is of Hopf type. 相似文献
2.
The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves,
which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling
waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four
envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability
boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify
longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of
the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of
behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries
separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion
system and to the weak electrolyte model for electroconvection in nematic liquid crystals.
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3.
Guillaume James 《Journal of Nonlinear Science》2012,22(5):813-848
We study the propagation of an unusual type of periodic travelling waves in chains of identical beads interacting via Hertz’s contact forces. Each bead periodically undergoes a compression phase followed by free flight, due to special properties of Hertzian interactions (fully nonlinear under compression and vanishing in the absence of contact). We prove the existence of such waves close to binary oscillations, and numerically continue these solutions when their wavelength is increased. In the long wave limit, we observe their convergence towards shock profiles consisting of small compression regions close to solitary waves, alternating with large domains of free flight where bead velocities are small. We give formal arguments to justify this asymptotic behavior, using a matching technique and previous results concerning solitary wave solutions. The numerical finding of such waves implies the existence of compactons, i.e. compactly supported compression waves propagating at a constant velocity, depending on the amplitude and width of the wave. The beads are stationary and separated by equal gaps outside the wave, and each bead reached by the wave is shifted by a finite distance during a finite time interval. Below a critical wave number, we observe fast instabilities of the periodic travelling waves, leading to a disordered regime. 相似文献
4.
Nikolai Bessonov Anne Beuter Sergei Trofimchuk Vitaly Volpert 《Mathematical Methods in the Applied Sciences》2019,42(11):3912-3928
A neural field model with different activation and inhibition connectivity and response functions is considered. Stability analysis of a homogeneous in space solution determines the conditions of the emergence of stationary periodic solutions and of periodic travelling waves. Various regimes of wave propagation are illustrated in numerical simulations. The influence of external stimulation on the wave properties is investigated. 相似文献
5.
The reaction-diffusion travelling waves that can be initiatedin an open isothermal chemical system governed by cubic autocatalytickinetics are discussed. The system is shown to be capable ofsustaining up to three spatially uniform steady states, the(trivial) unreacted state, which is always stable (a node),and two nontrivial states, one of which is always unstable (asaddle point). The third state can change its stability throughHopf bifurcation (both subcritical and supercritical). Thisallows the possibility of two sorts of travelling wave beingestablished; there are wave profiles which connect the unreactedstate ahead to the nontrivial state at the rear, and wave profiles(pulse waves) which have the unreacted state at both the frontand rear. The conditions under which a particular wave is initiatedare considered by both a discussion of the (ordinary) differentialequations governing the travelling waves and by numerical integrationsof an initial-value problem. This treatment also reveals thepossibility of a stable travelling wave propagating throughthe system, leaving behind a temporally unstable stationarystate. Under these conditions, spatiotemporal chaotic behaviouris seen to develop after the passage of the wave. 相似文献
6.
Shaolong Xie Lin Wang 《Communications in Nonlinear Science & Numerical Simulation》2012,17(3):1130-1141
In this paper, a generalized Camassa-Holm Kadomtsev-Petviashvili (gCH-KP) equation is studied. As a result, under different parameter conditions, the bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given, and the dynamic characters of these solutions are investigated. 相似文献
7.
Bifurcation of travelling wave solutions for (2+1)-dimension nonlinear dispersive long wave equation
In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed. 相似文献
8.
《Communications in Nonlinear Science & Numerical Simulation》2011,16(10):3941-3948
By using the dynamical system theory and the integral bifurcation method, a modified Camassa–Holm and Degasperis–Procession (CH–DP) equation are studied. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given, and the dynamic characters of these solutions are investigated. 相似文献
9.
Planar bifurcation method of dynamical system for investigating different kinds of bounded travelling wave solutions of a generalized Camassa-Holm equation
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In this study, by using planar bifurcation method of dynamical system, we study a generalized Camassa-Holm (gCH) equation. As results, under different parameter conditions, many bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given. The dynamic properties of these exact solutions are investigated. 相似文献
10.
《Applied Mathematics Letters》2006,19(5):458-463
In this work we demonstrate how the extension of the Evans function method using the compound matrix approach can be implemented to undertake the stability analysis (normally done through numerical means) of nonlinear travelling waves. The main advantage of this approach is that it can easily overcome the stiffness which is normally associated with these kinds of problems. We present a general approach which allows this method to be used for a general class of nonlinear travelling wave problems. 相似文献
11.
We study Klein–Gordon chains with attractive nearest neighbour forces and convex on-site potential, and show that there exists a two-parameter family of periodic travelling waves (wave trains) with unimodal and even profile functions. Our existence proof is based on a saddle-point problem with constraints and exploits the invariance properties of an improvement operator. Finally, we discuss the numerical computation of wave trains. 相似文献
12.
本文研究了广义特殊Tzitzeica-Dodd-Bullough类型方程,利用动力系统分支理论方法,证明该方程存在周期行波解,无界行波解和破切波解,并求出了一些用参数表示的显示精确行波解. 相似文献
13.
In this paper, we derive a time-periodic lattice model for a single species in a patchy environment, which has age structure and an infinite number of patches connected locally by diffusion. By appealing to the theory of asymptotic speed of propagation and monotonic periodic semiflows for travelling waves, we establish the existence of periodic travelling wave and spreading speed of the model. 相似文献
14.
Summary. In this paper, we consider the growth dynamics of a single-species population with two age classes and a fixed maturation
period living in a spatial transport field. A Reaction Advection Diffusion Equation (RADE) model with time delay and nonlocal
effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling waves
for the delay model with three birth functions which appeared in the well-known Nicholson's blowflies equation, and we consider
and analyze numerical solutions of the travelling wavefronts from the wave equations for the problems with nonlocal temporally
delayed effects. In particular, we report our numerical observations about the change of the monotonicity and the possible
occurrence of multihump waves. The stability of the travelling wavefront is numerically considered by computing the full time-dependent
partial differential equations with nonlocal delay. 相似文献
15.
16.
In this paper, we study some generalized Camassa-Holm equation. Through the analysis of the phase-portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave and compactons were discussed. In some certain parametric conditions, many exact solutions to the above travelling waves were given. Further-more, the 3D and 2D pictures of the above travelling wave solutions are drawn using Maple software. 相似文献
17.
BIFURCATIONS AND NEW EXACT TRAVELLING WAVE SOLUTIONS OF THE COUPLED NONLINEAR SCHRDINGER-KdV EQUATIONS
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By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrdinger-KdV equations are studied. Based on this method, all phase portraits of the system in the parametric space are given. All possible bounded travelling wave solutions such as solitary wave solutions and periodic travelling wave solutions are obtained. With the aid of Maple software, the numerical simulations are conducted for solitary wave solutions and periodic travelling wave solutions to the coupled nonlinear Schrdinger-KdV equations. The results show that the presented findings improve the related previous conclusions. 相似文献
18.
Mariana Haragus 《Studies in Applied Mathematics》2011,126(2):157-185
The Kadomtsev–Petviashvili (KP) equation possesses a four‐parameter family of one‐dimensional periodic traveling waves. We study the spectral stability of the waves with small amplitude with respect to two‐dimensional perturbations which are either periodic in the direction of propagation, with the same period as the one‐dimensional traveling wave, or nonperiodic (localized or bounded). We focus on the so‐called KP‐I equation (positive dispersion case), for which we show that these periodic waves are unstable with respect to both types of perturbations. Finally, we briefly discuss the KP‐II equation, for which we show that these periodic waves are spectrally stable with respect to perturbations that are periodic in the direction of propagation, and have long wavelengths in the transverse direction. 相似文献
19.
Zhou and Tian [J.B. Zhou, L.X. Tian, A type of bounded travelling wave solutions for the Fornberg-Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261] successfully found a type of bounded travelling wave solutions of the Fornberg-Whitham equation. In this paper, we improve the previous result by using the phase portrait analytical technology. Moreover, some smooth periodic wave, smooth solitary wave, periodic cusp wave and loop-soliton solutions are given, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation. 相似文献
20.
Jaime Angulo Pava 《Mathematische Nachrichten》2012,285(13):1572-1602
We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in $H^1_{per}$ . 相似文献