Dispersion engineering and enhanced light–matter interaction are the most desirable attributes for creating linear and nonlinear devices. It is always a challenging task to achieve both characteristics simultaneously. In this paper, we present a design of a silicon waveguide that provides not only enhanced light–matter interaction, measured through evanescent field strength, but also a platform to realize compact couplers and enhanced four-wave mixing efficiency. We make use of a convex-shaped design and numerically evaluate its modal characteristics. The design supports two zero dispersion wavelengths, from 1.2 µm to 1.3 µm and 1.7 µm to 1.8 µm, which can be tailored by changing the shape design parameters such as depth. Numerical simulations reveal that the depth of the convex shape affects both linear and nonlinear properties of the designed waveguide. We observe significant improvement of 15.2 dB in four-wave mixing efficiency when the waveguide is covered with a monolayer of graphene, illustrating the capabilities of such convex-shaped waveguides.