<p>This paper focuses on the existence, asymptotic behaviors and stability of slow-fast traveling pulses in a catalytic or electrochemical oscillation system with quintic nonlinearity. Inspired by the geometric singular perturbation theory and generalized rotated vector field, we provide the existence and wave speed for the traveling pulses with slow and fast dynamics induced by quintic nonlinearity with a parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$a\in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> in such chemical system. More precisely, we can provide the parameter interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a\in (\bar{a},1/\sqrt{3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msqrt> <mn>3</mn> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the occurrence of figure-eight type double homoclinic cycles corresponding to the bright and dark slow-fast traveling pulses, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\bar{a}=0.49616716645\cdots \pm \sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mo>=</mo> <mn>0.49616716645</mn> <mo>⋯</mo> <mo>±</mo> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> is a error parameter. Furthermore, the asymptotic behaviors of such slow-fast traveling pulses are exhibited by applying the asymptotic theory, and the nonlinear stability of slow-fast traveling pulses have been proved by the spectral theory.</p>

Existence and Stability of Slow-Fast Traveling Pulses in a Chemical System with Quintic Nonlinearity

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发表日期：

2025年2月19日

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Zhengkang Li,
Xingbo Liu

摘要

This paper focuses on the existence, asymptotic behaviors and stability of slow-fast traveling pulses in a catalytic or electrochemical oscillation system with quintic nonlinearity. Inspired by the geometric singular perturbation theory and generalized rotated vector field, we provide the existence and wave speed for the traveling pulses with slow and fast dynamics induced by quintic nonlinearity with a parameter $a\in [0,1]$ $a \in [0, 1]$ in such chemical system. More precisely, we can provide the parameter interval $a\in (\bar{a},1/\sqrt{3})$ $a \in (\bar{a}, 1 / \sqrt{3})$ for the occurrence of figure-eight type double homoclinic cycles corresponding to the bright and dark slow-fast traveling pulses, where $\bar{a}=0.49616716645\cdots \pm \sigma $ $\bar{a} = 0.49616716645 \dots \pm σ$ and $\sigma $ $σ$ is a error parameter. Furthermore, the asymptotic behaviors of such slow-fast traveling pulses are exhibited by applying the asymptotic theory, and the nonlinear stability of slow-fast traveling pulses have been proved by the spectral theory.