An efficient determination of critical parameters of nonlinear Schrödinger equation with a point-like potential using generalized polynomial chaos methods

We consider the nonlinear Schrödinger equation with a point-like source term. The soliton interaction with such a singular potential yields a critical solution behavior. That is, for the given value of the potential strength and the soliton amplitude, there exists a critical velocity of the initial soliton solution, around which the solution is either trapped by or transmitted through the potential. In this paper, we propose an efficient method for finding such a critical velocity by using the generalized polynomial chaos (gPC) method. For the proposed method, we assume that the soliton velocity is a random variable and expand the solution in the random space using orthogonal polynomials. Then the gPC method is used with spectral convergence. The proposed method is expected to find the critical velocity accurately with spectral convergence. Thus the computational complexity is much reduced. Numerical results for the smaller and higher values of the potential strength confirm the spectral convergence of the proposed method.