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Existence of isolated periodic waves of a family of PDEs with cubic reaction term

Krishna Patra, Ch. Srinivasa Rao

TL;DR

The paper addresses the existence of isolated periodic traveling waves in a family of reaction-convection-diffusion equations with cubic nonlinearity by reducing to a perturbed planar Hamiltonian system. It leverages Abelian integrals A(h)=\alpha_{0}A_{0}(h)+\alpha_{n}A_{n}(h) and the monotonicity of the ratio G_{n}(h)=A_{n}(h)/A_{0}(h) to bound the number of limit cycles bifurcating from periodic annuli. Depending on the parameter regimes for $u_{0}$ and $k$, the authors prove at most one such wave exists in several cases, while in others (notably $n$ odd with $u_{0}=-k$) no limit cycles occur. They also derive a procedure to determine intervals for the ratio $\alpha_{0}/\alpha_{n}$ that guarantee the existence of a limit cycle and corroborate the theory with numerical experiments, illustrating the practical impact for predicting isolated periodic traveling waves in ecological-type PDE models.

Abstract

In this paper we study the isolated periodic traveling wave solutions of a family of reaction-convection-diffusion equations with cubic reaction term. Existence/nonexistence of periodic traveling wave solutions are discussed in different parametric ranges. The monotonicity of the ratio of Abelian integrals is used to prove the existence of at most one limit cycle. Finally, numerical study is presented at the end.

Existence of isolated periodic waves of a family of PDEs with cubic reaction term

TL;DR

The paper addresses the existence of isolated periodic traveling waves in a family of reaction-convection-diffusion equations with cubic nonlinearity by reducing to a perturbed planar Hamiltonian system. It leverages Abelian integrals A(h)=\alpha_{0}A_{0}(h)+\alpha_{n}A_{n}(h) and the monotonicity of the ratio G_{n}(h)=A_{n}(h)/A_{0}(h) to bound the number of limit cycles bifurcating from periodic annuli. Depending on the parameter regimes for and , the authors prove at most one such wave exists in several cases, while in others (notably odd with ) no limit cycles occur. They also derive a procedure to determine intervals for the ratio that guarantee the existence of a limit cycle and corroborate the theory with numerical experiments, illustrating the practical impact for predicting isolated periodic traveling waves in ecological-type PDE models.

Abstract

In this paper we study the isolated periodic traveling wave solutions of a family of reaction-convection-diffusion equations with cubic reaction term. Existence/nonexistence of periodic traveling wave solutions are discussed in different parametric ranges. The monotonicity of the ratio of Abelian integrals is used to prove the existence of at most one limit cycle. Finally, numerical study is presented at the end.

Paper Structure

This paper contains 13 sections, 15 theorems, 70 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Classification of equilibrium points and phase portraits of the unperturbed system (\ref{['kpr1.17']})
  3. Case: $0<u_0<k$
  4. Case: $-k<u_0<0$
  5. Case: $u_0<-k$
  6. Limiting cases [$u_0=k$ and $u_0=-k$]
  7. Existence of isolated periodic waves of (\ref{['eq3.01']})
  8. Monotonicity of $G_{n}(h)$ for $0<u_0<k\leq2u_0$
  9. Monotonicity of $G_{n}(h)$ for the cases $-k<u_0<0$ and $u_0<-k$
  10. Monotonicity of $G_{n}(h)$ for $u_0=-k$
  11. Proof of Theorem \ref{['lemma3.01']}
  12. Interval of the ratio $\frac{\alpha_0}{\alpha_n}$ for the existence of limit cycle and numerical study
  13. Conclusion

Key Result

Theorem A

(Andronov et al.ref02 , Perkoref15) Let us consider the system of differential equations where $h(u)$ is an analytic function in a neighborhood of $u=u_0$, $h(u_0)=0,$$k \geq 2,$ and $\hat{\lambda} \neq 0.$ Then

Figures (14)

  • Figure 1: Phase portrait of the system (\ref{['kpr1.17']}) when $\beta=1,$$u_0=2,$ and $k=3.$
  • Figure 2: Phase portrait of the system (\ref{['kpr1.17']}) when $\beta=1,$$u_0=1,$ and $k=2.$
  • Figure 3: Phase portrait of the system (\ref{['kpr1.17']}) when $\beta=1,$$u_0=1,$ and $k=3.$
  • Figure 4: Phase portrait of the system (\ref{['kpr1.17']}) when $\beta=1,$$u_0=-1,$ and $k=2.$
  • Figure 5: Phase portrait of the system (\ref{['kpr1.17']}) when $\beta=1,$$u_0=-2,$ and $k=1.$
  • ...and 9 more figures

Theorems & Definitions (16)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 6 more