# Does a general soliton solution satisfy ALSO the normal wave equation?

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.

• And what about the 1-soliton kink solution from the sine-Gordon equation? Commented Feb 12, 2023 at 22:18
• The solitary wave of the s-G is also $f(x-vt)$ type so it will satisfy the 1D linear wave equation, and also asymptotically its multi-soliton solution, see en.wikipedia.org/wiki/Sine-Gordon_equation Commented Feb 12, 2023 at 22:24
• So, multisolitons do NOT generally satisfy the normal wave equations but 1-solitons do, right? Commented Feb 12, 2023 at 23:35
• while in or near collision it is a big mess, only the asymptotic satisfies the 1D linear WE. Commented Feb 12, 2023 at 23:57