A quantitative study of the nonlinear Schrِdinger equation with singular potential of any derivative orders

In this letter, we report, for the first time, the quantitative study of the nonlinear Schrödinger equation with the singular potential term represented by the derivative of the Dirac δ -function of higher orders, δ ( n ) , n ≥ 1 , via numerical approximation. We found that a similar critical phenomenon occurs with δ ( n ) as in the case with the δ -function. That is, the soliton solution is split into transmitted, trapped, and reflected solutions and the transmitted and reflected parts preserve the soliton structure. Furthermore, the higher derivative of the impulsive forcing term is used the stronger reflection occurs; the reflection coefficient increases almost exponentially as the order increases. We also found that for each order the reflection coefficient decays almost exponentially with the soliton velocity. The decay pattern of the trapping rates for higher values of n is different from that with n = 0 if n is large. Various velocities with up to fifth order derivative of the δ -function are used to verify the claim.