The application of the “inverse problem” method for constructing confining potentials that make N-soliton waveforms exact solutions in the Gross–Pitaevskii equation

Abstract

In this work, we discuss an application of the “inverse problem” method to find the external trapping potential, which has particular N trapped soliton-like solutions of the Gross–Pitaevskii equation (GPE) also known as the cubic nonlinear Schrödinger equation (NLSE). This inverse method assumes particular forms for the trapped soliton wave function, which then determines the (unique) external (confining) potential. The latter renders these assumed waveforms exact solutions of the GPE (NLSE) for both attractive (g<0) and repulsive (g>0) self-interactions. For both signs of g, we discuss the stability with respect to self-similar deformations and translations. For g<0, a critical mass Mc or equivalently the number of particles for instabilities to arise can often be found analytically. On the other hand, for the case with g>0 corresponding to repulsive self-interactions which is often discussed in the atomic physics realm of Bose–Einstein condensates, the bound solutions are found to be always stable. For g<0, we also determine the critical mass numerically by using linear stability or Bogoliubov–de Gennes analysis, and compare these results with our analytic estimates. Various analytic forms for the trapped N-soliton solutions in one, two, and three spatial dimensions are discussed, including sums of Gaussians or higher-order eigenfunctions of the harmonic oscillator Hamiltonian.