Soliton Solutions Associated With a Class of Third-Order Ordinary Linear Differential Operators

Explicit solutions to the related integrable nonlinear evolution equations are constructed by solving the inverse scattering problem in the reflectionless case for the third-order differential equation $d^3\psi /d{x}^3+Qd\psi /dx+P\psi =k^3\psi$, where $Q$ and $P$ are the potentials in the Schwartz class and $k^3$ is the spectral parameter. The input data set used to solve the relevant inverse problem consists of the bound-state poles of a transmission coefficient and the corresponding bound-state dependency constants. Using the time-evolved dependency constants, explicit solutions to the related integrable evolution equations are obtained. In the special cases of the Sawada–Kotera equation and the modified bad Boussinesq equation, the method presented here explains the physical origin of the constants appearing in the relevant $N$-soliton solutions algebraically constructed, but without any physical insight, by the bilinear method of Hirota.