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In this article, we have constructed some new soliton solutions of the nonlocal nonlinear medium equation via three reliable approaches. The executed approaches are modified Kudryashov method, sinh-Gordon equation expansion method and extended sinh-Gordon equation expansion method. In this case, there are three types of competing nonlinearities that are taken into account in our model. They are nonlocal nonlinearity, cubic nonlinearity, and quintic nonlinearity. By means of the aforementioned methods, dark, bright, combined dark-bright, singular, combined singular, periodic, and other soliton solutions are obtained from the nonlocal nonlinear medium equation, and their respective existence conditions.

Optical Soliton in Nonlocal Nonlinear Medium with Cubic-Quintic Nonlinearities and Spatio-Temporal Dispersion