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The rotation modified Kadomtsev Petviashvili equation which is also known as the Kadomtsev Petviashvili Ostrovsky equation, describes the gradual wave field diffusion in the transverse direction to the direction of the propagation of the wave in a rotating frame of reference. This equation is a generalization of the Ostrovsky equation additionally having weak transverse effects. We investigate transverse instability and stability of small periodic traveling waves of the Ostrovsky equation with respect to either periodic or square-integrable perturbations in the direction of wave propagation and periodic perturbations in the transverse direction of motion in the rotation-modified Kadomtsev Petviashvili equation. We also study transverse stability or instability in generalized rotation modified KP equation by taking dispersion term as general and quadratic and cubic nonlinearity. As a consequence, we obtain transverse stability or instability in two dimensional generalization of RMBO eqution, Ostrovsky Gardner equation, Ostrovsky fKdV equation, Ostrovsky mKdV equation, Ostrovsky ILW equation, Ostrovsky Whitham etc.

We study the modulational instability of small-amplitude periodic traveling wave solutions in a dispersion generalized Ostrovsky equation. Specifically, we investigate the invertibility of the associated linearized operator in the vicinity of the origin and derive a modulational instability index that depends on the dispersion and nonlinearity. For the classical Ostrovsky equation, we recover the well-known Lighthill condition for modulational instability of small-amplitude periodic traveling waves, and further provide a rigorous connection of the Lighthill condition to the spectral instability of the underlying wave. Our results and methodologies further apply to a wide-class of Ostrovsky type models that incorporate various dispersive effects. As such, we present new results illuminating the effects of rotation on various full-dispersion models arising in the study of weakly nonlinear surface water waves., Comment: 29 pages, 11 figures. Correcting typos, adding comparison to Lighthill condition for modulational instability of small periodic waves

The tropical cyclone formation process is one of the most complex natural phenomena which is governed by various atmospheric, oceanographic, and geographic factors that varies with time and space. Despite several years of research, accurately predicting tropical cyclone formation remains a challenging task. While the existing numerical models have inherent limitations, the machine learning models fail to capture the spatial and temporal dimensions of the causal factors behind TC formation. In this study, a deep learning model has been proposed that can forecast the formation of a tropical cyclone with a lead time of up to 60 hours with high accuracy. The model uses the high-resolution reanalysis data ERA5 (ECMWF reanalysis 5th generation), and best track data IBTrACS (International Best Track Archive for Climate Stewardship) to forecast tropical cyclone formation in six ocean basins of the world. For 60 hours lead time the models achieve an accuracy in the range of 86.9% - 92.9% across the six ocean basins. The model takes about 5-15 minutes of training time depending on the ocean basin, and the amount of data used and can predict within seconds, thereby making it suitable for real-life usage.

We study the transverse spectral stability of the one-dimensional small-amplitude periodic traveling wave solutions of the (2+1)-dimensional Konopelchenko-Dubrovsky (KD) equation. We show that these waves are transversely unstable with respect to two-dimensional perturbations that are periodic in both directions with long wavelength in the transverse direction. We also show that these waves are transversely stable with respect to perturbations which are either mean-zero periodic or square-integrable in the direction of the propagation of the wave and periodic in the transverse direction with finite or short wavelength. We discuss the implications of these results for special cases of the KD equation - namely, KP-II and mKP-II equations.