Eye of the tyger: Early-time resonances and singularities in the inviscid Burgers equation

We chart a singular landscape in the temporal domain of the inviscid Burgers equation in one space dimension for sine-wave initial conditions. These so far undetected complex singularities are arranged in an eye shape centered around the origin in time. Interestingly, since the eye is squashed along the imaginary-time axis, complex-time singularities can become physically relevant at times well before the first real singularity-the preshock. Indeed, employing a time-Taylor representation for the velocity around t = 0, loss of convergence occurs roughly at 2/3 of the preshock time for the considered single-and mul-timode models. Furthermore, the loss of convergence is accompanied by the appearance of initially localized resonant behavior which, as we claim, is a temporal manifestation of the so-called tyger phenomenon, reported in Galerkin-truncated implementations of inviscid fluids [S. S. Ray et al., Phys. Rev. E 84, 016301 (2011)]. We support our findings of early-time tygers with two complementary and independent means, namely, by an asymptotic analysis of the time-Taylor series for the velocity, as well as by a singularity theory that employs Lagrangian coordinates. Finally, we apply two methods that reduce the amplitude of early-time tygers. One is tyger purging which removes large Fourier modes from the velocity, and is a variant of a procedure known in the literature. The other method realizes an iterative UV completion, which, most interestingly, iteratively restores the conservation of energy once the Taylor series for the velocity diverges. Our techniques are straightforwardly adapted to higher dimensions and/or applied to other equations of hydrodynamics.

Automated summary

This study explores the complex singularities in the temporal domain of the inviscid Burgers equation with sine-wave initial conditions, revealing a unique eye-shaped arrangement of singularities centered around the origin. These complex-time singularities become physically relevant before the preshock, showcasing the importance of early-time phenomena in fluid dynamics. Various methods are employed to reduce the amplitude of these early-time singularities, including tyger purging and iterative UV completion, demonstrating potential applications in higher dimensions and other hydrodynamic equations.