Is the word "essentially" an adjective in phrase "essentially nonlinear phenomena"?

No, it is an adverb, modifying the adjective essential.

This means we say things like:

We will discuss essentially nonlinear phenomena

but

Will will discuss the essential nonlinearity of that system.

Essentially may be seen as an adverb, modifying are/is. It modifies the manner of being.

In this case if we wrote “These are essentially nonlinear phenomena” we would mean that the manner of being of the phenomena is that they are nonlinear.

Such a statement in a mathematical or scientific context draws attention to the fundamental nature of the phenomena. The use of essentially implies that they are in essence nonlinear: that is to say that nonlinearity lies at the heart of their being and defines their nature. Nonlinearity is not merely lightly descriptive but is essential to their nature.

Essential = basic to the nature of someone or something

Cambridge dictionary