共查询到20条相似文献,搜索用时 250 毫秒
1.
Wenzhang Huang 《Journal of Dynamics and Differential Equations》2001,13(1):147-183
For a system of reaction–diffusion equations that models the interaction of n mutualist species, the existence of the bistable traveling wave solution has been proved where the nonlinear reaction terms possess a certain type of monotonicity. However the problem of whether there can be two distinct traveling waves remains open. In this paper we use a homotopy approach incorporated with the Liapunov–Schmidt method to show that the bistable traveling wave solution is unique. Our method developed in this paper can also be applied to study the existence and uniqueness of traveling wave solutions for some competition models. 相似文献
2.
Jonatan Lenells 《Journal of Dynamics and Differential Equations》2006,18(2):381-391
We classify the weak traveling wave solutions for a class of one-dimensional non-linear shallow water wave models. The equations are shown to admit smooth, peaked, and cusped solutions, as well as more exotic waves such as stumpons and composite waves. We also explain how some previously studied traveling wave solutions of the models fit into this classification. 相似文献
3.
One-dimensional traveling wave solutions for imbibition processes into a homogeneous porous medium are found within a recent generalized theory of macroscopic capillarity. The generalized theory is based on the hydrodynamic differences between percolating and nonpercolating fluid parts. The traveling wave solutions are obtained using a dynamical systems approach. An exhaustive study of all smooth traveling wave solutions for primary and secondary imbibition processes is reported here. It is made possible by introducing two novel methods of reduced graphical representation. In the first method the integration constant of the dynamical system is related graphically to the boundary data and the wave velocity. In the second representation the wave velocity is plotted as a function of the boundary data. Each of these two graphical representations provides an exhaustive overview over all one-dimensional and smooth solutions of traveling wave type, that can arise in primary and secondary imbibition. Analogous representations are possible for other systems, solution classes, and processes. 相似文献
4.
Zhehao Huang Zhengrong Liu Zhenzhen Wang 《Journal of Dynamics and Differential Equations》2016,28(2):389-417
In this paper, we consider the stochastic KPP equation which is perturbed by an environmental noise. Applying an extended stochastic ordering technique, we establish the existence of a stochastic traveling wave solution to the equation and give a sufficient condition under which solutions can be attracted to the stochastic traveling wave. 相似文献
5.
Cheng-Hsiung Hsu Jian-Jhong Lin Tzi-Sheng Yang 《Journal of Dynamics and Differential Equations》2017,29(1):323-342
This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation. 相似文献
6.
We study traveling wavefront solutions for two reaction–diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument. 相似文献
7.
Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis.For any given initial data with two pieces of constant states,the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states,the solution may be constructed from the Riemann solutions,with each two adjacent states connected by elementary waves.A new Riemann problem forms when these two waves collide.Through the exploration of these Riemann problems,the outcome of wave interactions may be classified in a suitable parametric space. 相似文献
8.
In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution. 相似文献
9.
In this paper we investigate traveling wave solutions of a non-linear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal. To describe the response of viscoelastic solids we assume a non-linear relationship among the linearized strain, the strain rate and the Cauchy stress. We then concentrate on traveling wave solutions that correspond to the heteroclinic connections between the two constant states. We establish conditions for the existence of such solutions, and find those solutions, explicitly, implicitly or numerically, for various forms of the non-linear constitutive relation. 相似文献
10.
11.
Weiguo Rui 《Nonlinear dynamics》2014,76(2):1529-1542
It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations. 相似文献
12.
Wenzhang Huang 《Journal of Dynamics and Differential Equations》2012,24(3):633-644
We use a shooting method to show the existence of traveling wave fronts and to obtain an explicit expression of minimum wave speed for a class of diffusive predator?Cprey systems. The existence of traveling wave fronts indicates the existence of a transition zone from a boundary equilibrium to a co-existence steady state and the minimum wave speed measures the asymptotic speed of population spread in some sense. Our approach is a significant improvement of techniques introduced by Dunbar. The advantage of our method is that it does not need the notion of Wazewski??s set and LaSalle??s invariance principle used in Dunbar??s approach. In our approach, we convert the equations for traveling wave solutions to a system of first order equations by a ??non-traditional transformation??. With this converted new system, we are able to construct a Liapunov function, which gives an immediate implication of the boundedness and convergence of the relevant class of heteroclinic orbits. Our method provides a more efficient way to study the existence of traveling wave solutions for general predator?Cprey systems. 相似文献
13.
In this paper, the existence of periodic traveling wave solutions with a priori unknown velocity is considered for a coupled map lattice dynamical system. By trasforming our problem into one that involves polynomials, explicit 2- and 3-periodic traveling wave solutions are found, while the other solutions can be computed numerically. Since there does not seem to be any reports on explicit traveling wave solutions, we hope that our results will lead to the discovery of many others. 相似文献
14.
IntroductionHowtoobtaintravelingwavespeedsandsolutionsinnonlinearreaction_diffusionequationshasbeenaclassoffocusedquestionsformathematiciansandtheoreticalphysicists.Lineardiffusionyieldsinfinitepropagationspeed ,sodiffusioncoefficientswhichdependonthe… 相似文献
15.
We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave. 相似文献
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17.
Chang-Hong Wu 《Journal of Dynamics and Differential Equations》2016,28(2):317-338
In this paper, we develop a general approach to deal with the asymptotic behavior of traveling wave solutions in a class of three-component lattice dynamical systems. Then we demonstrate an application of these results to construct entire solutions which behave as two traveling wave fronts moving towards each other from both sides of x-axis for a three-species competition system with Lotka–Volterra type nonlinearity in a lattice. 相似文献
18.
IntroductionCamassa ,Holm[1]obtainedaclassofnewcompletelyintegrableshallowwaterequation ,i.e.,Camassa_Holmequation2ut+ 2kux-12 uxxt+ 6uux =uxuxxx+ 12 uuxxx. ( 1 )Foreveryk,theEq .( 1 )isaclassofcompletelyintegrablesystem .Thisclassofequationisaclassofnotonlystrangebutalso… 相似文献
19.
Wenxian Shen 《Journal of Dynamics and Differential Equations》2011,23(1):1-44
The current paper is devoted to the study of traveling wave solutions of spatially homogeneous monostable reaction diffusion
equations with ergodic or recurrent time dependence, which includes periodic and almost periodic time dependence as special
cases. Such an equation has two spatially homogeneous and time recurrent solutions with one of them being stable and the other
being unstable. Traveling wave solutions are a type of entire solutions connecting the two spatially homogeneous and time
recurrent solutions. Recently, the author of the current paper proved that a spatially homogeneous time almost periodic monostable
equation has a spreading speed in any given direction. This result can be easily extended to monostable equations with recurrent
time dependence. In this paper, we introduce generalized traveling wave solutions for time recurrent monostable equations
and show the existence of such solutions in any given direction with average propagating speed greater than or equal to the
spreading speed in that direction and non-existence of such solutions of slower average propagating speed. We also show the
uniqueness and stability of generalized traveling wave solutions in any given direction with average propagating speed greater
than the spreading speed in that direction. Moreover, we show that a generalized traveling wave solution in a given direction
with average propagating speed greater than the spreading speed in that direction is unique ergodic in the sense that its
wave profile and wave speed are unique ergodic, and if the time dependence of the monostable equation is almost periodic,
it is almost periodic in the sense that its wave profile and wave speed are almost periodic. 相似文献
20.
Ying Huang 《Nonlinear dynamics》2013,72(1-2):87-90
With the aid of the known Bäcklund transformation, starting from some given traveling solutions, we consider new exact no-traveling wave solutions to the Liouville equation, and a series of breather soliton solutions, doubly periodic solutions, two-soliton solutions as well as periodic-soliton solutions are obtained. 相似文献