Eur. Phys. J. D (2025) 79: 149
https://doi.org/10.1140/epjd/s10053-025-01096-6

Research - Photons

Dynamics of nonlinear wave of the (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation with variable coefficients in inhomogeneous media

¹ Department of Mathematics, Faculty of Science, Soran University, Erbil, Iraq
² Department of Mathematics, College of Science, University of Duhok, Duhok, Iraq

^a This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 7 October 2025
Accepted: 12 November 2025
Published online: 8 December 2025

Abstract

This study investigates nonlinear wave propagation in inhomogeneous media governed by the (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation with time-dependent variable coefficients. Employing the Hirota bilinear method combined with the long wave limit technique, we systematically construct exact analytical solutions including multi-soliton, lump wave, breather wave, and their hybrid interaction solutions. Unlike prior studies limited to constant-coefficient formulations, we provide the first systematic analysis of how variable coefficient functions $g(t)\in \{csc(t),sec(t),cosh(t)\}$ fundamentally alter wave dynamics. Our results reveal that periodic coefficients such as $csc(t)$ and $sec(t)$ induce temporal modulation and resonance phenomena in soliton interactions, while exponential coefficients like $cosh(t)$ drive amplitude amplification mechanisms critical for rogue wave formation. We demonstrate that hybrid wave interactions in inhomogeneous media exhibit qualitatively distinct collision dynamics compared to homogeneous cases, including energy redistribution, phase-dependent scattering, and temporal gating effects. These findings directly inform dispersion-managed optical fiber systems where time-varying group velocity dispersion controls pulse propagation, shallow water wave modeling over variable-depth ocean floors where topography-induced inhomogeneity governs tsunami dynamics, and plasma physics applications involving ion-acoustic waves in non-uniform density profiles. The systematic framework established here enables quantitative prediction of nonlinear wave behavior in realistic non-uniform media, advancing both theoretical understanding and practical applications in wave control and nonlinear signal processing.

Graphical abstract: This graphical abstract illustrates the rich nonlinear wave dynamics of the (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation with variable coefficients in inhomogeneous media. The study employs the Hirota bilinear method combined with the long wave limit technique to derive multiple classes of exact analytical solutions, including multi-soliton, lump, breather, and hybrid interaction structures. The influence of variable coefficients such as $g(t)=\text{csc}(t),\text{sec}(t)$, and $\text{cosh}(t)$ on the wave propagation and interaction patterns is analyzed in depth. The results reveal that time-dependent inhomogeneity strongly affects the amplitude, width, and stability of nonlinear structures, providing key insights into energy localization and transfer in optical, fluid, and plasma systems. Graphical Summary: $\bullet$ Multi-soliton, lump, and breather waves visualized through 2D/3D surface and contour plots. $\bullet$ Interaction phenomena including soliton–lump, soliton–breather, and lump–breather dynamics. $\bullet$ Variable coefficients modulate the nonlinear behavior, revealing tunable propagation properties. $\bullet$ Potential applications in nonlinear optics, plasma waves, and dispersive fluid systems.