Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi–Pasta–Ulam–Tsingou recurrence

Abstract

Modulation instability, breather formation, and the Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A “cascading mechanism” is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with “eye-shaped” breathers in one waveguide and “four-petal” modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.

Superposition of modulated nonlinear waves in inhomogeneous systems with negative coherent coupling

“…Considering the interactions of different field components, coupled models are employed in many physical sciences. In optical systems, the coherently coupled NLS (CNLS) equations [11][12][13][14][15][16][17][18][19][20][21][22][23][24] have been demonstrated to be useful in investigating the dynamic of orthogonal polarization waves in weakly birefringent fibers, and in many areas of optical communication, such as the collision-based logic and wavelength-division-multiplexed transmission. The coherent coupling may cause the energy exchange between the orthogonal polarization pulses and results to the richer characteristics of nonlinear waves [16,17].…”

Construction of Novel Bright-Dark Solitons and Breather Waves of Unstable Nonlinear Schrödinger Equations with Applications

“…These equations are crucial tools for comprehending the physical analogies and variations in the nonlinear behaviour of dispersive waves. Rogue waves in nonlinear dispersive media are very thought provoking, for example, [16][17][18][19][20] . These rogue waves were initially examined in the background of water waves, and after that, they became an interesting topic for in-depth studies of nonlinear optics.…”