$O(2)$-Hopf 分岔处的变分非线性与波选择
Variational Nonlinearities and Wave Selection at an $O(2)$-Hopf Bifurcation
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- Marmara University（马尔马拉大学）
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中文总结 AI 辅助
本文证明在 $O(2)$-Hopf 分岔中，变分非线性结构强制三次系数实部消失，从而决定行波与驻波的稳定性选择。
中文摘要 AI 辅助
在 $O(2)$ 等变 Hopf 分岔处，两个三次正规形系数 $\zeta_R$ 和 $\xi_R$ 的实部控制着行波与驻波之间的选择。在四阶正则化 $p$-系统中，观察到 $\xi_R$ 消失且三次非线性仅移动频率。我们证明这两个观察都是结构性的：即使线性部分是耗散且非哈密顿的，变分非线性也强制它们成立。我们考虑圆上一类双分量偏微分方程，其二次和三次非线性由场的反射不变局部哈密顿量及其有限多个空间导数通过一个公共常系数 Poisson 算子生成。那么对于每个可容许的线性部分都有 $\xi_R=0$，而 $\zeta_R$ 仅依赖于二次非线性。对于变分非线性，直接三次项是纯虚的。混合级联项失去其实部，因为其二次谐波具有零时间频率且预解式保持其奇偶性，而在加倍临界频率处的自级联保持实部。当 $\zeta_R\neq 0$ 时，分岔的行波是鞍点，而驻波在 $\zeta_R<0$ 时是超临界且轨道渐近稳定的，在 $\zeta_R>0$ 时是次临界且不稳定的。当非线性是 Poisson 算子作用于仅含场的多项式，且其符号在临界和二次谐波波数处不为零时，每个可容许线性部分的相消反过来刻画了变分性。在耗散正则化的 Boussinesq 族中，当耗散趋于零时，$\zeta_R$ 在共振外线性消失，在 2:1 共振处反比发散。因此三次系数的哈密顿极限是奇异的。
英文摘要
At an $O(2)$-equivariant Hopf bifurcation, the real parts $ζ_R$ and $ξ_R$ of the two cubic normal-form coefficients govern selection between traveling and standing waves. In a fourth-order regularized $p$-system, $ξ_R$ was observed to vanish and the cubic nonlinearity to shift only frequencies. We show that both observations are structural: variational nonlinearities enforce them even when the linear part is dissipative and non-Hamiltonian. We consider a class of two-component PDEs on the circle whose quadratic and cubic nonlinearities are generated by reflection-invariant local Hamiltonians of the fields and finitely many of their spatial derivatives, through a common constant-coefficient Poisson operator. Then $ξ_R=0$ for every admissible linear part, while $ζ_R$ depends only on the quadratic nonlinearity. For variational nonlinearities the direct cubic terms are purely imaginary. The mixed cascade term loses its real part because its second harmonic has zero temporal frequency and the resolvent keeps its parity, while the self-cascade at the doubled critical frequency keeps a real part. When $ζ_R\ne0$, the bifurcating traveling waves are saddles, while the standing waves are supercritical and orbitally asymptotically stable for $ζ_R<0$ and subcritical and unstable for $ζ_R>0$. When the nonlinearities are the Poisson operator applied to polynomials in the fields alone, and its symbol does not vanish at the critical and second-harmonic wavenumbers, cancellation for every admissible linear part conversely characterizes variationality. In a dissipatively regularized Boussinesq family, as dissipation tends to zero, $ζ_R$ vanishes linearly off resonance and diverges inversely at the 2:1 resonance. The Hamiltonian limit of the cubic coefficient is therefore singular.
