The formation dynamics of periodic ripples induced by femtosecond laser pulses (pulse duration τ = 50 fs and central wavelength λ = 800 nm) are studied by a collinear pump-probe imaging technique with a temporal resolution of 1 ps and a spatial resolution of 440 nm. The ripples with periods close to the laser wavelength begin to appear upon irradiation of two pump pulses at surface defects produced by the prior one. The rudiments of periodic ripples emerge in the initial tens of picoseconds after fs laser irradiation, and the ripple positions keep unmoved until the formation processes complete mainly in a temporal span of 1500 ps. The results suggest that the periodic deposition of laser energy during the interaction between femtosecond laser pulses and sample surface plays a dominant role in the formation of periodic ripples.

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The Peregrine soliton is an exact, rational, and localized solution of the nonlinear Schrödinger equation and is commonly employed as a model for rogue waves in physical sciences. If the transverse variable is allowed to be complex by analytic continuation while the propagation variable remains real, the poles of the Peregrine soliton travel down and up the imaginary axis in the complex plane. At the turning point of the pole trajectory, the real part of the complex variable coincides with the location of maximum height of the rogue wave in physical space. This feature is conjectured to hold for at least a few other members of the hierarchy of Schrödinger equations. In particular, evolution systems with coherent coupling or quintic (fifth-order) nonlinearity will be studied. Analytical and numerical results confirm the validity of this conjecture for the first-and second-order rogue waves.