AI 中文总结
针对周期Sobolev空间上的一类非局部非线性项，通过将其重新表述为自由薛定谔算子生成的酉群下实解析函数的共振轨道平均，证明其可表示为受迫薛定谔方程系统温和解的迹，并将该结果应用于非局部振幅方程的等价局部重新表述。
AI 中文摘要
我们证明，周期Sobolev空间上的一类非局部非线性项可表示为由受迫薛定谔方程系统构成的局部色散问题的唯一温和解的迹。该证明基于将非局部非线性项重新表述为自由薛定谔算子$\boldsymbol{i} \boldsymbol{\nabla}_x^2$生成的酉群下实解析函数的共振轨道平均。我们通过为新近获得的非局部振幅方程提供等价的局部重新表述来说明该结果，该方程形式上捕捉了守恒Hopf不稳定性附近抛物系统的动力学。
英文摘要
We prove that a class of nonlocal nonlinearities on periodic Sobolev spaces can be expressed as a trace of the unique mild solution to a local dispersive problem consisting of a system of forced Schrödinger equations. The proof is based on a reformulation of the nonlocal nonlinearities as the resonant orbit average of a real-analytic function under the unitary group generated by the free Schrödinger operator $\mathrm{i} \partial_x^2$. We illustrate our result by providing an equivalent local reformulation for a recently obtained nonlocal amplitude equation that formally captures the dynamics of parabolic systems close to a conserved Hopf instability.
Comments9 pages
