Appearance of bound states in random potentials with applications to soliton theory

We analyze the stochastic creation of a single bound state (BS) in a random potential with a compact support. We study both the Hermitian Schrödinger equation and non-Hermitian Zakharov-Shabat systems. These problems are of special interest in the inverse scattering method for Korteveg–de-Vries and the nonlinear Schrödinger equations since soliton solutions of these two equations correspond to the BSs of the two aforementioned linear eigenvalue problems. Analytical expressions for the average width of the potential required for the creation of the first BS are given in the approximation of delta-correlated Gaussian potential and additionally different scenarios of eigenvalue creation are discussed for the non-Hermitian case.

Figure 1

The phase portrait of Eq. (6).

Figure 2

The scaled average length of the potential required for creating a level below $E$ vs. the dimensionless parameter $\gamma$.

Figure 3

The scaled average length of potential required for creating a level above $\eta$ vs. the dimensionless parameter $\eta /D$.