非线性薛定谔方程参数共振的异宿结构
Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation
摘要：
我们证明了在{ \em defocusing}非线性薛定谔方程中由参量驱动引起的调制不稳定性的非线性阶段可以通过结合模式截断和平均方法来精确描述，在强驱动区域是有效的.由此产生的可积振子揭示了一个复杂的隐藏的异宿结构的不稳定性。一个显着的结果，验证了原始模型的数值积分，是呼吸解决方案的存在，分离不同的费米-Pasta-Ulam经常性制度。我们的理论还表明，最佳参数放大意外地发生在线性化Floquet分析所产生的共振（或阿诺德舌头）的带宽之外。
英文摘要：
We show that the nonlinear stage of modulational instability induced by parametric driving in the {\em defocusing} nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis.
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