The Biswas–Mollivic equation is a special type of nonlinear Schrödinger equation, which explains the spatio-temporal behaviour of excitable media. In this paper, we investigate the optical soliton solutions of the Biswas–Mollivic equation with cubic–quintic–septic–nonic nonlinearities using the generalized Riccati equation mapping method. This method is efficient and provides new perspectives. It also provides novel insights into the dynamics of excitable media. Our findings add to a better understanding of the complex spatio-temporal patterns that develop in excitable media and have potential applications in the design of new technologies for controlling and manipulating pattern formation. To depict optical soliton solutions graphically, we use the MATLAB software.

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified $$ \left( \dfrac{G'}{G^2}\right) $$G ′ G 2 -expansion and improved $$\tan \left( \dfrac{\phi \left( \xi \right) }{2}\right) $$tan ϕ ξ 2 -expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified $$ \left( \dfrac{G'}{G^2}\right) $$G ′ G 2 -expansion and improved $$\tan \left( \dfrac{\phi \left( \xi \right) }{2}\right) $$tan ϕ ξ 2 -expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches.