Breathers, modulation instability and recurrence phenomena are studied for the derivative nonlinear Schrödinger equation, which incorporates second order dispersion, cubic nonlinearity and self-steepening effect. By insisting on periodic boundary conditions, a cascading process will occur where initially small higher order Fourier modes can grow alongside with lower order modes. Typically a breather is first observed when all modes attain roughly the same order of magnitude. Beyond the formation of the first breather, analytical formula of spatially periodic but temporally localized breather ceases to be a valid indicator. However, numerical simulations display Fermi–Pasta–Ulam–Tsingou type recurrence. Self-steepening effect plays a crucial role in the dynamics, as it induces motion of the breather and generates chaotic behavior of the Fourier coefficients. Theoretically, correlation between breather motion and the Lax pair formulation is made. Physically, quantitative assessments of wave profile evolution are made for different initial conditions, e.g. random noise versus modulation instability mode of maximum growth rate. Potential application to fluid mechanics is discussed.

研究了导数非线性薛定谔方程的呼吸、调制不稳定性和递归现象，该方程结合了二阶色散、三次非线性和自陡峭效应。通过坚持周期性边界条件，将发生级联过程，其中最初小的高阶傅立叶模式可以与低阶模式一起增长。通常，当所有模式达到大致相同的数量级时，首先会观察到喘息。除了第一个呼吸器的形成之外，空间周期性但时间上局部化的呼吸器的分析公式不再是一个有效的指标。然而，数值模拟显示了 Fermi-Pasta-Ulam-Tsingou 型重现。自陡效应在动力学中起着至关重要的作用，因为它会引起呼吸器的运动并产生傅立叶系数的混沌行为。理论上，呼吸运动与松散对公式之间存在相关性。在物理上，对不同初始条件（例如随机噪声与最大增长率的调制不稳定性模式）进行波形演变的定量评估。讨论了流体力学的潜在应用。