Transition waves for lattice Fisher-KPP equations with time and space dependence

2020 ◽  
pp. 1-28
Author(s):  
Ning Wang ◽  
Zhi-Cheng Wang ◽  
Xiongxiong Bao
Keyword(s):  
Wave Solution ◽  
Critical Speed ◽  
Wave Speed ◽  
Existence Results ◽  
Kpp Equations ◽  
Space Dependence ◽  
Time Periodic ◽  
Subsolutions And Supersolutions ◽  
Heterogeneous Lattice ◽  
Transition Wave

Abstract This paper is concerned with the existence results for generalized transition waves of space periodic and time heterogeneous lattice Fisher-KPP equations. By constructing appropriate subsolutions and supersolutions, we show that there is a critical wave speed such that a transition wave solution exists as soon as the least mean of wave speed is above this critical speed. Moreover, the critical speed we construct is proved to be minimal in some particular cases, such as space-time periodic or space independent.

COEXISTENCE OF MULTIFARIOUS EXACT NONLINEAR WAVE SOLUTIONS FOR GENERALIZED b-EQUATION

2010 ◽  
Vol 20 (10) ◽  
pp. 3193-3208 ◽  
Author(s):  
RUI LIU
Keyword(s):  
Nonlinear Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Solitary Wave Solution ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Wave Solutions ◽  
Special Cases ◽  
Singular Wave ◽  
Nonlinear Wave Solutions

In this paper, we consider the generalized b-equation ut - uxxt + (b + 1)u2ux = buxuxx + uxxx. For a given constant wave speed, we investigate the coexistence of multifarious exact nonlinear wave solutions including smooth solitary wave solution, peakon wave solution, smooth periodic wave solution, single singular wave solution and periodic singular wave solution. Not only is the coexistence shown, but the concrete expressions are given via phase analysis and special integrals. From our work, it can be seen that the types of exact nonlinear wave solutions of the generalized b-equation are more than that of the b-equation. Many previous results are turned to our special cases. Also, some conjectures and questions are presented.

scholarly journals Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ning Wang ◽  
Zhi-Cheng Wang
Keyword(s):  
Diffusion Model ◽  
Critical Speed ◽  
Wave Speed ◽  
Upper And Lower Solutions ◽  
Reaction Diffusion ◽  
Linear Operators ◽  
Time Space ◽  
Reaction Diffusion Model ◽  
Periodic Reaction ◽  
Asymptotic Speed

<p style='text-indent:20px;'>This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.</p>

THE EXPLICIT NONLINEAR WAVE SOLUTIONS AND THEIR BIFURCATIONS OF THE GENERALIZED CAMASSA–HOLM EQUATION

2011 ◽  
Vol 21 (11) ◽  
pp. 3119-3136 ◽  
Author(s):  
ZHENGRONG LIU ◽  
YONG LIANG
Keyword(s):  
Solitary Wave ◽  
Nonlinear Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Solitary Wave Solution ◽  
Wave Solutions ◽  
Holm Equation ◽  
Bifurcation Phenomena ◽  
Singular Wave ◽  
Nonlinear Wave Solutions

In this paper, we study the explicit nonlinear wave solutions and their bifurcations of the generalized Camassa–Holm equation [Formula: see text]Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed.Firstly, it is verified that k = 3/8 is a bifurcation parametric value for several types of explicit nonlinear wave solutions.When k < 3/8, there are five types of explicit nonlinear wave solutions, which are(i) hyperbolic peakon wave solution,(ii) fractional peakon wave solution,(iii) fractional singular wave solution,(iv) hyperbolic singular wave solution,(v) hyperbolic smooth solitary wave solution.When k = 3/8, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.When k > 3/8, there is not any type of explicit nonlinear wave solutions.Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.

scholarly journals Travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Chloé Colson ◽  
Faustino Sánchez-Garduño ◽  
Helen M. Byrne ◽  
Philip K. Maini ◽  
Tommaso Lorenzi
Keyword(s):  
Wave Solution ◽  
Degradation Rate ◽  
Wave Speed ◽  
Propagation Speed ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Tumour Invasion ◽  
Wave Analysis ◽  
Shooting Argument ◽  
Minimal Wave Speed

In this paper, we carry out a travelling-wave analysis of a model of tumour invasion with degenerate, cross-dependent diffusion. We consider two types of invasive fronts of tumour tissue into extracellular matrix (ECM), which represents healthy tissue. These types differ according to whether the density of ECM far ahead of the wave front is maximal or not. In the former case, we use a shooting argument to prove that there exists a unique travelling-wave solution for any positive propagation speed. In the latter case, we further develop this argument to prove that there exists a unique travelling-wave solution for any propagation speed greater than or equal to a strictly positive minimal wave speed. Using a combination of analytical and numerical results, we conjecture that the minimal wave speed depends monotonically on the degradation rate of ECM by tumour cells and the ECM density far ahead of the front.

Novel Weissenberg effects

1977 ◽  
Vol 81 (2) ◽  
pp. 265-272 ◽  
Author(s):  
G. S. Beavers ◽  
D. D. Joseph
Keyword(s):  
Critical Speed ◽  
Angular Frequency ◽  
Low Frequencies ◽  
Integral Number ◽  
Weissenberg Effect ◽  
Stress Amplification ◽  
The Mean ◽  
Viscoelastic Liquids ◽  
Time Periodic ◽  
Good Agreement

We have observed two novel manifestations of the Weissenberg effect in viscoelastic liquids which are set into motion by the rotation of a circular rod. In the first experiment we floated a layer of STP on water. The STP climbs up the rod into the air and down the rod into the water. The ‘down-climb’ is much larger than the ‘up-climb’, their ratio being roughly the square root of the density difference (STP-air)/ (water–STP). The magnification of the down-climb may be regarded asnormal-stress amplification. [dagger] The magnitudes of the up- and down-climbs are simultaneously in good agreement with the predictions of a theory of rod climbing when the angular frequency of the rod is small. In the second experiment, we set the rod into torsional oscillations. When the amplitude of the oscillation is small, the fluid climbs the rod; the climb is divided into an axisymmetric steady mean part and an oscillating part (Joseph 1976b; Beavers 1976). The mean axisymmetric climb dominates the total climb at low frequencies. At a higher critical speed the axisymmetric climbing bubble loses its stability to another time-periodic motion with the same period but with a ‘flower’ pattern displaying a certain integral number of petals.

Travelling wave solutions of a parabolic-hyperbolic system for contact inhibition of cell-growth

2015 ◽  
Vol 26 (3) ◽  
pp. 297-323 ◽  
Author(s):  
M. BERTSCH ◽  
D. HILHORST ◽  
H. IZUHARA ◽  
M. MIMURA ◽  
T. WAKASA
Keyword(s):  
Partial Differential Equations ◽  
Cell Growth ◽  
Large Class ◽  
Wave Solution ◽  
Wave Speed ◽  
Large Time ◽  
Travelling Wave ◽  
Travelling Wave Solution ◽  
Travelling Wave Solutions ◽  
Wave Solutions

We consider a cell growth model involving a nonlinear system of partial differential equations which describes the growth of two types of cell populations with contact inhibition. Numerical experiments show that there is a parameter regime where, for a large class of initial data, the large time behaviour of the solutions is described by a segregated travelling wave solution with positive wave speed c. Here, the word segregated expresses the fact that the different types of cells are spatially segregated, and that the single densities are discontinuous at the moving interface which separates the two populations. In this paper, we show that, for each wave speed c > c, there exists an overlapping travelling wave solution, whose profile is continuous and no longer segregated. We also show that, for a large class of initial functions, the overlapping travelling wave solutions cannot represent the large time profile of the solutions of the system of partial differential equations. The structure of the travelling wave solutions strongly resembles that of the scalar Fisher-KPP equation, for which the special role played by the travelling wave solution with minimal speed has been extensively studied.

Cylindrically symmetric travelling fronts in a periodic reaction—diffusion equation with bistable nonlinearity

2015 ◽  
Vol 145 (5) ◽  
pp. 1053-1090 ◽  
Author(s):  
Zhi-Cheng Wang
Keyword(s):  
Wave Speed ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Qualitative Properties ◽  
One Dimensional ◽  
Cylindrically Symmetric ◽  
Periodic Reaction ◽  
Bistable Nonlinearity ◽  
Travelling Fronts ◽  
Time Periodic

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

scholarly journals Traveling waves in cooperative predation: relaxation of sublinearity

2021 ◽  
pp. 1-10
Author(s):  
Srijana Ghimire ◽  
Xiang-Sheng Wang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Traveling Wave Solutions ◽  
Traveling Wave Solution ◽  
Positive Equilibrium ◽  
Open Problems ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Wave Solutions

In this paper, we investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation. Sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. On the other hand, if the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. In the end of this paper, we propose five open problems related to traveling wave solutions in cooperative predation.

Sign in / Sign up

Export Citation Format

Share Document