scholarly journals Travelling wave solutions in a negative nonlinear diffusion–reaction model

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1495-1522
Author(s):  
Yifei Li ◽  
Peter van Heijster ◽  
Robert Marangell ◽  
Matthew J. Simpson
Keyword(s):  
Nonlinear Diffusion ◽  
Wave Speed ◽  
Geometric Approach ◽  
Travelling Wave ◽  
Reaction Model ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Minimum Wave Speed ◽  
Nonlinear Diffusion Reaction Equation

AbstractWe use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

scholarly journals Propagation of stochastic travelling waves of cooperative systems with noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hao Wen ◽  
Jianhua Huang ◽  
Yuhong Li
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Wave Speed ◽  
Travelling Waves ◽  
Equilibrium Points ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Cooperative System ◽  
Wave Solutions ◽  
Suitable Marker

<p style='text-indent:20px;'>We consider the cooperative system driven by a multiplicative It\^o type white noise. The existence and their approximations of the travelling wave solutions are proven. With a moderately strong noise, the travelling wave solutions are constricted by choosing a suitable marker of wavefront. Moreover, the stochastic Feynman-Kac formula, sup-solution, sub-solution and equilibrium points of the dynamical system corresponding to the stochastic cooperative system are utilized to estimate the asymptotic wave speed, which is closely related to the white noise.</p>

scholarly journals Superharmonic instability of nonlinear travelling wave solutions in Hamiltonian systems

2019 ◽  
Vol 876 ◽  
pp. 896-911 ◽  
Author(s):  
Naoki Sato ◽  
Michio Yamada
Keyword(s):  
Hamiltonian System ◽  
Wave Speed ◽  
Hamiltonian Dynamics ◽  
Travelling Wave ◽  
Linear Instability ◽  
Wave Profile ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Present Argument ◽  
Exchange Of Stability

The problem of linear instability of a nonlinear travelling wave in a canonical Hamiltonian system with translational symmetry subject to superharmonic perturbations is discussed. It is shown that exchange of stability occurs when energy is stationary as a function of wave speed. This generalizes a result proved by Saffman (J. Fluid Mech., vol. 159, 1985, pp. 169–174) for travelling wave solutions exhibiting a wave profile with reflectional symmetry. The present argument remains true for any non-canonical Hamiltonian system that can be cast in Darboux form, i.e. a canonical Hamiltonian form on a submanifold defined by constraints, such as a two-dimensional surface wave on a constant shearing flow, revealing a general feature of Hamiltonian dynamics.

scholarly journals Diffusive waves in inhomogeneous media

1989 ◽  
Vol 32 (2) ◽  
pp. 291-315 ◽  
Author(s):  
Paul C. Fife
Keyword(s):  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Monotone Function ◽  
Travelling Wave ◽  
Inhomogeneous Media ◽  
Nonlinear Diffusion Equation ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Stable Solutions ◽  
Image Position

When the function f(u) is of “bistable type’, i.e. has two zeros h̲ and h+ at which f' is negative and (for simplicity) has only one other zero between them, then the constant functions u = h± are L∞-stable solutions of the nonlinear diffusion equationIn addition, there are travelling wave solutions u+(x, t) and u̲(x, t) which, ifconnect h+ to h̲ in the sense thatthe convergence being uniform on bounded x-intervals. These solutions are of the formwhere U(z) is a monotone function (the wave's profile), U(±∞) = h±, and the velocity c is a specific positive number depending on the function f.

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.

Spreading speeds and traveling wave solutions of diffusive vector-borne disease models without monotonicity

2021 ◽  
pp. 1-30
Author(s):  
Xinjian Wang ◽  
Guo Lin ◽  
Shigui Ruan
Keyword(s):  
Wave Speed ◽  
Reproduction Number ◽  
Travelling Wave ◽  
Spreading Speed ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Kinetic System ◽  
Minimal Wave Speed ◽  
Vector Borne ◽  
The Basic Reproduction Number

Vector-borne diseases, such as chikungunya, dengue, malaria, West Nile virus, yellow fever and Zika, pose a major global public health problem worldwide. In this paper we investigate the propagation dynamics of diffusive vector-borne disease models in the whole space, which characterize the spatial expansion of the infected hosts and infected vectors. Due to the lack of monotonicity, the comparison principle cannot be applied directly to this system. We determine the spreading speed and minimal wave speed when the basic reproduction number of the corresponding kinetic system is larger than one. The spreading speed is mainly estimated by the uniform persistence argument and generalized principal eigenvalue. We also show that solutions converge locally uniformly to the positive equilibrium by employing two auxiliary monotone systems. Moreover, it is proven that the spreading speed is the minimal wave speed of travelling wave solutions. In particular, the uniqueness and monotonicity of travelling waves are obtained. When the basic reproduction number of the corresponding kinetic system is not larger than one, it is shown that solutions approach to the disease-free equilibrium uniformly and there is no travelling wave solutions. Finally, numerical simulations are presented to illustrate the analytical results.

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