I checked that the usual wave funtions of a gaussian pulse, a $\text{sech}(x-vt)$ and $\text{sech}^2(x-vt)$ solitons (the two latter from KdV equations) satisfy the wave equation.
Is this general? I mean, are every travelling wave solutions a solution of the usual linear wave equation in addition to non-linear equations where they "truly" arise?
Given a density function $\rho:\mathbb R^n \to \mathbb R$ one can write a soliton as $$\phi(\mathbf r, t) = \rho(\mathbf r-\mathbf v t)$$although whether it actually solves some other differential equation depends on the specifics of how you choose $\rho$ and $\mathbf v$.
Due to this form, any such soliton satisfies the chain-rule equations, $${\partial \phi\over\partial t} = -\mathbf v\cdot \nabla \rho=-\mathbf v\cdot\nabla\phi$$ (the unidirectional wave equation), and $${\partial^2 \phi\over\partial t^2} = (\mathbf v\cdot \mathbf v)\nabla^2\phi$$ (the isotropic wave equation), which can be helpful in various contexts.
Since the basic solitary wave solution of the KdV has the form of $f(x-vt)$ for some $f(\cdot)$ it also solves the 1-D wave equation $\frac{\partial^2 f}{\partial x^2}-\frac{1}{v^2}\frac{\partial^2 f}{\partial x^2}=0$. More general soliton solutions, ones that start at $-\infty$, scatter here, and move onto $+\infty$ in their separate ways are also combination of $f(x-vt)$ types so they, too, satisfy the 1-D wave equation. The reverse statement is not true.
