I am studying the following Lamé equation in the Jacobi form \begin{equation} \frac{d^2 v}{dx^2}  \left(2k^2 \operatorname{cn}^{2}\left [x\;\;k\right ]\right)v = \lambda v, \end{equation} where $k\in(0, 1)$ is parameter related to the elliptic modulus of the Jacobi elliptic functions. In order to better understand this equation I am first taking the soliton limit when $k\rightarrow 1$ to obtain the following differential equation \begin{equation} \frac{d^2 v}{dx^2}  2\operatorname{sech}^2{(x)}v = \lambda v, \end{equation} for which analytic solutions exist depending on the values of the parameter $\lambda$. For different behaviour of eigenfunction $v$ is observed for $\lambda=1,$ $\lambda\in(\infty,1)\cup(1,0)$ and $\lambda\geq 0.$ I believe I obtained the correct solutions; however, I am having hard time finding them explicitly in the literature. Does anyone know good resources on the latter equation, or if it is known under a special name? My ultimate goal is to use these solutions in the context of the KdV equation, which is a compatibility condition for the stationary Schrödinger equation \begin{equation}\label{SE} \frac{d^2 v}{dx^2}  u(x,t) v = \lambda v \end{equation}
1 Answer
The Schrödinger equation with the $\text{sech}^2$ potential (sometimes called the Pöschl–Teller potential) was first studied by Epstein in 1930 [1]. There is an extensive literature on this exactly solvable case. Two recent references are [2,3]. A special feature of this potential is that it is reflectionless, it admits unit transmission at all energies. For that reason it has also found many realworld applications, in particular in photonics.
 P.S. Epstein, Reflection of waves in an inhomogeneous absorbing medium (1930).
 J. Lekner, Reflectionless eigenstates of the $\text{sech}^2$ potential (2007).
 C.S. Park, Transmission time of a particle in the reflectionless sechsquared potential (2011).

2$\begingroup$ A further nice pedagogical paper on the reflectionless potentials, solved via supersymmetry operators, is physics.smu.edu/scalise/P6335fa19/notes/… $\endgroup$– BuzzJun 26 at 5:56