New analytical wave structures of the (3+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3+1)$$\\end{document}-dimensional extended modified Ito equation of seventh-order

Abstract

Partial differential equations are frequently employed to depict issues arising across various scientific and engineering domains. Efforts have been made to analytically solve these equations, revealing shortcomings in some widely utilized methods, including modeling deficiencies and intricate solution processes. To address these limitations, diverse analytical methods have been explored. The Ito equation, introduced in 1980, underwent development, leading to the formulation of a fifth-order Ito equation. A seventh-order integrable (3+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3+1)$$\\end{document}-dimensional extended modified Ito equation emerged by augmenting this equation with three additional terms. In this study, novel exact solutions for the equation, absent in existing literature, were derived using the extended hyperbolic function and modified Kudryashov methods. To scrutinize the dynamic behavior of these findings, we presented 3D, contour, and 2D visualizations of select solutions. The results showcase numerous new solutions, underscoring the reliability and efficacy of the employed methods.