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In this work, we show the application of the ``inverse problem'' method toconstruct exact $N$ trapped soliton-like solutions of the nonlinearSchr\"odinger or Gross-Pitaevskii equation (NLSE and GPE, respectively) in one,two, and three spatial dimensions. This method is capable of finding theexternal (confining) potentials which render specific assumed waveforms exactsolutions of the NLSE for both attractive ($g<0$) and repulsive ($g>0$)self-interactions. For both signs of $g$, we discuss the stability with respectto self-similar deformations and translations. For $g<0$, a critical mass$M_c$, or equivalently the number of particles, for instabilities to arise canoften be found analytically. On the other hand, for the case with $g>0$corresponding to repulsive self interactions which is often discussed in theatomic physics realm of Bose-Einstein condensates (BEC), the bound solutionsare found to be always stable. For $g<0$, we also determine the critical massnumerically by using linear stability or Bogoliubov-de Gennes analysis, andcompare these results with our analytic estimates. Various analytic forms forthe trapped $N$-soliton solutions are discussed, including sums of Gaussians orhigher-order eigenfunctions of the harmonic oscillator Hamiltonian.

Exact trapped $N$-soliton solutions of the nonlinear Schr\"odinger equation using the inverse problem method