<p>In this paper, we consider the <i>b</i>-family of the Novikov equations with cubic nonlinearities, which serves as a one-parameter family of Camassa-Holm type equations and includes the integrable Novikov equation in the case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$b=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We establish the spectral instability of the periodic peaked waves to this family in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^2({\mathbb {T}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^1({\mathbb {T}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, utilizing the method of characteristics, we investigate the relation between the time evolution of the linearized equation and the spectral properties of the linearized operator in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^2({\mathbb {T}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. </p>

Spectral instability of the periodic peaked waves for the b-family of the Novikov equations

期刊论文
发表日期：

2026年6月15日

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Haijing Song,
Ying Fu

摘要

In this paper, we consider the b-family of the Novikov equations with cubic nonlinearities, which serves as a one-parameter family of Camassa-Holm type equations and includes the integrable Novikov equation in the case $b=3$ $b = 3$ . We establish the spectral instability of the periodic peaked waves to this family in $L^2({\mathbb {T}})$ $L^{2} (T)$ and $H^1({\mathbb {T}})$ $H^{1} (T)$ . Moreover, utilizing the method of characteristics, we investigate the relation between the time evolution of the linearized equation and the spectral properties of the linearized operator in $L^2({\mathbb {T}})$ $L^{2} (T)$ .