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In my professor's lecture notes, I came across the next approach of studying non-linear waves in fluids. So we have compressible Navier-Stokes and continuity equation in 1D and we assume adiabatic ideal gas. After linearizing them about some stable point ($\vec{u}_0 = 0, \rho_0, p_o $), we get linear waves equations and we get the expression for the speed of sound: $$ c = \sqrt{\frac{\partial{p}}{\partial{\rho}}}$$ Then, he defines some $P$ as $ dP = \frac{dp}{\rho c}$. Then, after doing some rewriting and rearranging one can easily obtain from the Navier-Stokes and conservation of mass two equations: $$ \left[ \frac{\partial{}}{\partial{t}}+(u+c)\frac{\partial{}}{\partial{x}}\right](u+P)=0$$ $$ \left[ \frac{\partial{}}{\partial{t}}+(u-c)\frac{\partial{}}{\partial{x}}\right](u-P)=0$$ He then states that one can write down the implicit solution for a nonlinear sound wave propagating in some direction and see that solution can show that a shock wave can form in the fluid but does not go into much more detail about this approach. Has anyone seen this approach before? Is there a name for it? What do those equations tell us and is this an actually valid approach to non-linear waves or is this too simplistic? Can someone point me towards some further reading on this?

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Has anyone seen this approach before?

Yes, the term that leads to a shock is the $u \ \tfrac{\partial u}{\partial x}$ term, which in general form looks like $\mathbf{u} \cdot \nabla \mathbf{u}$. This term describes the nonlinear steepening (e.g., see my discussion in the following answer https://physics.stackexchange.com/a/139436/59023) of a sound wave. This just shows that for a wave whose phase speed depends upon amplitude, the leading front will decrease in spatial scale, i.e., it will steepen.

Thus, the equations you presented show that for a linear system in the absence of energy dissipation, sound waves will steepen without bound and reach a gradient catastrophe. If a gradient catastrophe occurs, then the wave will break much like large water waves (e.g., sometimes called white caps or breakers).

The part that is currently missing is some dissipative term, i.e., some form of energy dissipation that can balance the nonlinear steepening. This is required for shock initiation. What is generally assumed is that the dissipative terms are incredibly small until the wave front steepens to some critical level, at which point something like viscosity kicks in and limits further steepening, i.e., a term like $\mu \ \nabla^{2} \mathbf{u}$.

Is there a name for it?

Not sure if there is a name for it other than the typical introduction to sound wave steepening and shock formation.

What do those equations tell us and is this an actually valid approach to non-linear waves or is this too simplistic?

For an upper limit of a "weak shock," the two equations you present are perfectly valid assuming the dissipative term is present but neglected until the gradient scale length of the steepened sound wave becomes comparable to the mean free path of the particles for a collision-dominated fluid. Once that point is reached, then a term like $\mu \ \nabla^{2} \mathbf{u}$ will start to become important and can limit further increase in $\mathbf{u} \cdot \nabla \mathbf{u}$.

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