Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique

Abstract The present study investigates different types of wave symmetries in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>3</m:mn> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(3+1) -dimensional Chafee–Infante equation via the Hirota bilinear transformation technique. In this work, we derived exact solutions that include bright and dark solitons, periodic cross kink, multiple waves, mixed waves, homoclinic breathers, M-shaped related waves and periodic lump waves. These solutions exhibit stability, energy confinement, and dynamic interactions. It is observed that the Hirota method captures the highly nonlinear complex phenomena that result from the balance between the nonlinearity and the dispersion. These factors lead to a stability and coherent formation of wave forms. We employed Mathematica 11.1 software to obtain 3D, contour, and 2D graphs of our solution. The graphs present the spatial and temporal evolution of these solutions. The periodic structures, oscillatory solitons, and cross-kink configurations have dynamic interaction while maintaining fundamental properties of waves. Breather and homoclinic breather solutions present the basis of oscillatory local dynamics, which stress on energy transfer and phase modulation. The novelty of this article is in the use of the Hirota bilinear technique to the Chafee–Infante equation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>3</m:mn> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(3+1) -dimensional dimension, which allows the derivation of a wide variety of exact multidimensional wave solutions, including intricate hybrid solutions previously unreported for the equation. This provides great value by extending the analytical theory and improving the understanding of nonlinear wave behaviors in high-dimensional environments. It is worth noting that all the solutions have been verified and found to satisfy the governing equation.