Soliton

Remarks

The second clause is often checked by the existence of exact multi-soliton solutions, which split for $t \to \pm \infty$ into separated solitary waves with the same speeds. There is no single formal definition used by all authors.

Example: for the Korteweg–De Vries equation $u_t + 6 u u_x + u_{xxx} = 0$ and $c > 0$, put $\varphi(\xi) = \frac{c}{2} \operatorname{sech}^2\big(\frac{\sqrt{c}}{2} \xi\big)$, using the hyperbolic secant. For $u = \varphi(x - ct)$ the equation becomes $-c\varphi' + 6\varphi\varphi' + \varphi''' = 0$, which integrates to $\varphi'' = c\varphi - 3\varphi^2$. With $\varphi = A \operatorname{sech}^2(k\xi)$ one has $\varphi'' = 4k^2 A \operatorname{sech}^2(k\xi) - 6k^2 A \operatorname{sech}^4(k\xi)$, so the equation holds with $4k^2 = c$ and $A = 2k^2 = c/2$. The KdV equation also has explicit two-soliton solutions, so this solitary wave is a soliton.

Non-example: the linear equation $u_t + u_{xxx} = 0$ has no solitary wave with $c \ne 0$ that decays at infinity. Indeed, $\varphi''' = c\varphi'$ gives $\varphi'' = c\varphi$, whose decaying solutions on all of $\R$ are only $\varphi = 0$.