Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete Classification of Traveling Waves and Resolution of Linear Conjecture in Monostable Systems

Changhong Wu, Dongyuan Xiao, Maolin Zhou

Abstract

In this paper, we present a complete classification of traveling wave solutions for monostable systems within a unified framework. To this end, we introduce a novel technique, referred to as the slicing method, which is based on the construction of suitable super- and sub-solutions.

Complete Classification of Traveling Waves and Resolution of Linear Conjecture in Monostable Systems

Abstract

In this paper, we present a complete classification of traveling wave solutions for monostable systems within a unified framework. To this end, we introduce a novel technique, referred to as the slicing method, which is based on the construction of suitable super- and sub-solutions.
Paper Structure (33 sections, 29 theorems, 597 equations, 11 figures)

This paper contains 33 sections, 29 theorems, 597 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Traveling waves of three typical monostable systems
  3. Intuitive explanation on the slicing method
  4. Outline of the paper
  5. Main results
  6. The scalar reaction-diffusion equation
  7. The scalar nonlocal equation
  8. The Lotka-Volterra competition system
  9. Threshold of the reaction-diffusion equation
  10. Construction of the super-solution
  11. Proof of Theorem \ref{['th: threshold scalar equation']}
  12. Threshold of the nonlocal diffusion equation
  13. Preliminary
  14. Construction of the super-solution
  15. Proof of Theorem \ref{['th: threshold scalar equation nonlocal']}
  16. ...and 18 more sections

Key Result

Proposition 1.2

Assume $f(\cdot)$ satisfies the monostable condition monostable cd. The traveling wavefronts $(c,W)$, defined as in def of scalar tw, satisfies Here, $\lambda^{\pm}(c)$ are defined as

Figures (11)

  • Figure 2.1: The transition from linear selection to nonlinear selection of \ref{['sample eq']}.
  • Figure 2.2: The horizontal axis is the time; the vertical axis represents $x(t)/t$; the orange line indicates the value $2\sqrt{1-a}=\sqrt{2}$, and the blue curve represents the evolution of $x(t)/t$ on different $d$.
  • Figure 2.3: the blue curve represents the evolution of $x(t)/t$ on different $r$.
  • Figure 3.1: the construction of $R_w(\xi)$.
  • Figure 4.1: the construction of $\mathcal{R}_w(\xi)$.
  • ...and 6 more figures

Theorems & Definitions (80)

  • Remark 1.1
  • Proposition 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • ...and 70 more