This work deals with the dynamics of higher-order rogue waves in a new integrable (2+1)-dimensional Boussinesq equation governing the evolution of high and steep gravity water waves. To achieve this objective, we construct rogue wave solutions by employing Bell polynomial and Hirota’s bilinearization method, along with the generalized polynomial function. Through the obtained rogue wave solutions, we explore the impact of various system and solution parameters in their dynamics. Primarily, these parameters determine the characteristics of rogue waves, including the identification of their type, bright or dark type doubly-localized rogue wave structures and spatially localized rational solitons, and manipulation of their amplitude, depth, and width. Reported results will be encouraging to the studies on the rogue waves in higher dimensional systems as well as to experimental investigations on the controlling mechanism of rogue waves in optical systems, atomic condensates, and deep water oceanic waves.

This article investigates the dust ion acoustic multi-shock wave excitations in weakly relativistic multi-component plasma by assuming nonthermal, nonextensive electrons and positrons, relativistic ion fluid having kinetic viscosity and immobile dust. Burgers equation is derived to investigate such excitations by applying the reductive perturbation method. The exponential functions are directly implemented to determine the novel multi-shock wave solution of Burgers equation. The dust ion acoustic (DIA) multi-shock wave excitations are investigated systematically to reveal the effects of parameters, namely, viscosity coefficient of ions, positron to electron density ratio, immobile dust to electron density ratio, ion to electron temperature ratio, electron to positron temperature ratio, and relativistic streaming factor of ions in the presence of nonthermal, nonextensive, and concurrently acting nonthermal and nonextensive electrons as well as positrons. It is found that the amplitudes and widths of not only single, but also multi-shock wave compressive and rarefactive electrostatic potential structures are changed with the influence of all plasma parameters. The obtained results may be useful to analyze the nature of DIA multi-shock wave phenomena in various astrophysical as well as space environments (particularly, in pulsar relativistic winds with supernova ejecta) and future studies in plasma laboratory.

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Considering the importance of ever-increasing interest in exploring localized waves, we investigate a generalized (3+1)-dimensional Hirota-Satsuma-Ito equation describing the unidirectional propagation of shallow-water waves and perform Painlevé analysis to understand its integrability nature. We construct the explicit form of higher-order rogue wave solutions by adopting Hirota’s bilinearization and generalized polynomial functions. Further, we explore their dynamics in detail, depicting different pattern formation that reveal potential advantages with available arbitrary constants in their manipulation mechanism. Particularly, we demonstrate the existence of singly-localized line-rogue waves and doubly-localized rogue waves with multiple (single, triple, and sextuple) structures generating triangular and pentagon type geometrical patterns with controllable orientations that can be altered appropriately by tuning the parameters. The presented analysis will be an essential inclusion in the context of rogue waves in higher-dimensional systems.