Breathers, cascading instabilities and Fermi-Pasta-Ulam-Tsingou recurrence of the derivative nonlinear Schrodinger equation: Effects of 'self-steepening' nonlinearity

Breathers, modulation instability and recurrence phenomena are studied for the derivative nonlinear Schrodinger 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. (c) 2021 Elsevier B.V. All rights reserved.

Automated summary

Breathers, modulation instability, and recurrence phenomena are studied in the derivative nonlinear Schrodinger equation with second order dispersion, cubic nonlinearity, and self-steepening effect. Numerical simulations and theoretical analysis reveal the significant role of self-steepening effect in the dynamics and chaotic behavior of Fourier coefficients.