Within Gauss–Bonnet gravity, we construct a solution endowed with dyonic matter fields in a higher dimension. The quasi-topological electromagnetism generates two kinds of contributions; one is the kinetic terms, and the second refers to the interactif terms. This overcomes the invariance topological problem. We investigate the thermodynamical proprieties of the obtained solution, namely, ADM mass, Hawking temperature, and entropy. To inspect the local stability, we examine the associated heat capacity. With regards to optical proprieties, we analyze the null geodesic in terms of the given parameter space. The shadow radius is a generating form with all the physical parameters that govern the shadow behavior. The study restricts only the taking of the effects of the D and $$\alpha $$ α parameters. Finally, we examine the impact of the dimension D, GB coupling constant $$\alpha $$ α , the cosmological constant $$\Lambda $$ Λ , the electric $$q_e$$ q e , the magnetic charge $$q_m$$ q m and the coupling constant $$\beta $$ β on the energy emission rate.

Given the general nonlinear partial differential equations and the importance of the Korteweg-de Vries equation (KdV) in physics, this chapter presents a basic survey of the two-dimensional Korteweg-de Vries model. We begin by examining various symmetries of systems, and then explore the concept of integrability through two different methods: the Hamiltonian formalism and the existence of conserved quantities. By introducing the concept of q-deformation, we construct the corresponding q-deformation integrable model and the integrability of the resulting system is guaranteed by the existence of Lax pairs. We also present the KdV equation in the Moyal space of moments in its noncommutative version, we present the algebraic structure of the system and we study the integrability using the notion of Lax pair.

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