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We obtain the spectrum of bound states for a modified P\"oschl-Teller andsquare potential wells in the nonlinear Schr\"odinger equation. For a fixednorm of bound states, the spectrum for both potentials turns out to consist ofa finite number of multi-node localized states. We use modulational instabilityanalysis to derive the relation that gives the number of possible localizedstates and the maximum number of nodes in terms of the width of the potential.Soliton scattering by these two potentials confirmed the existence of thelocalized states which form as trapped modes. Critical speed for quantumreflection was calculated using the energies of the trapped modes.

Bound states spectrum of the nonlinear Schr\"odinger equation with P\"oschl-Teller and square potential wells