Analytical solutions and implications for hydrodynamics and nonlinear optics

This article discusses the generalization of time-fractional conformable mathematical models for studying the Estevez–Mansfield–Clarkson (EMC) equation of solutions. The numerical evaluation of liquid drop evolution, as studied through mathematical physics and nonlinear optics, uses the EMC equation in the integral-order model. Through the Kumar–Malik approach, scientists can identify analytical solutions for EMC problems that employ nonlinear fractional partial differential equations. Engineers can use first-order differential equation models to produce solutions that exhibit varied combinations between bright and dark features as well as periodic wave configurations of singular solutions. The inclusion of specific equation elements makes a system produce traveling wave solutions along with bright solitons and dark solitons. The investigation generated function-based solutions featuring exponential, trigonometric, hyperbolic, and Jacobi elliptic expressions. The experimental changes applied produce different patterns of wave solutions. The analytical work yielded discoveries regarding hydrodynamic wave physics and nonlinear optics together with plasma phenomena through its documentation of parameter-sensitive wave distribution reactions. Through this study, scientists pushed EMC equations into physical application areas and developed essential tools to advance future investigations in those domains. The analysis of complex physical systems requires scientists to follow multiple methods for developing analytical EMC equation solutions alongside graphical representations.