Explicit internal wave solutions in nonlinear fluid models on the whole space

We present explicit solutions to incompressible Euler and Navier-Stokes equations on $\mathbb{R}^n$, as well as the rotating Boussinesq equations on $\mathbb{R}^3$. These solutions are superpositions of certain linear waves of arbitrary amplitudes that also solve the nonlinear equations by constraints on wave-direction and wave-vectors. For $n\leq 3$ these are explicit examples for generalised Beltrami flows. We show that forcing terms of corresponding wave-type yield explicit solutions by linear variation of constants. We work in eulerian coordinates and distinguish the two situations of vanishing nonlinearity and of gradient nonlinearity, where the nonlinear term modifies the pressure. The methods introduced here for finding explicit solutions can also be used in other equations with material derivative.