Quadratic Peregrine solitons resonantly radiating without higher-order dispersion

Resonant radiation emitted by solitary waves via cascading in quadratic media

We present a systematic investigation of the resonant radiation emitted by localized soliton-like wave-packets supported by second-harmonic generation in the cascading regime. We emphasize a general mechanism which allows for the resonant radiation to grow without the need for higher-order dispersion, primarily driven by the second-harmonic component, while radiation is also shed around the fundamental-frequency component through parametric down-conversion processes. The ubiquity of such a mechanism is revealed with reference to different localized waves such as bright solitons (both fundamental and second-order), Akhmediev breathers, and dark solitons. A simple phase matching condition is put forward to account for the frequencies radiated around such solitons, which agrees well with numerical simulations performed against changes of material parameters (say, phase mismatch, dispersion ratio). The results provide explicit understanding of the mechanism of soliton radiation in quadratic nonlinear media.

Explicit Soliton Structure Formation for the Riemann Wave Equation and a Sensitive Demonstration

The motive of the study was to explore the nonlinear Riemann wave equation, which describes the tsunami and tidal waves in the sea and homogeneous and stationary media. This study establishes the framework for the analytical solutions to the Riemann wave equation using the new extended direct algebraic method. As a result, the soliton patterns of the Riemann wave equation have been successfully illustrated, with exact solutions offered by the plane solution, trigonometry solution, mixed hyperbolic solution, mixed periodic and periodic solutions, shock solution, mixed singular solution, mixed trigonometric solution, mixed shock single solution, complex soliton shock solution, singular solution, and shock wave solutions. Graphical visualization is provided of the results with suitable values of the involved parameters by Mathematica. It was visualized that the velocity of the soliton and the wave number controls the behavior of the soliton. We are confident that our research will assist physicists in predicting new notions in mathematical physics.

Spatiotemporal doubly periodic waves in a phase-mismatched second-harmonic generation

In this Letter, we present an analytical and numerical investigation to characterize the formation of quadratic doubly periodic waves originating from coherent modulation instability in a dispersive quadratic medium in the regime of cascading second-harmonic generation. To the best of our knowledge, such an endeavor has not been undertaken before, despite the growing relevance of doubly periodic solutions as the precursor of highly localized wave structures. Unlike the case with cubic nonlinearity, the periodicity of quadratic nonlinear waves can also be controlled by the wave-vector mismatch in addition to the initial input condition. Our results may impact widely on the formation, excitation, and control of extreme rogue waves and the description of modulation instability in a quadratic optical medium.