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In this work, we consider a ``reverse-engineering'' approach to constructconfining potentials that support exact, constant density kovaton solutions tothe classical Gross-Pitaevskii equation (GPE) also known as the nonlinearSchr\"odinger equation (NLSE). In the one-dimensional case, the exact solutionis the sum of stationary kink and anti-kink solutions, i.e. a kovaton, and inthe overlapping region, the density is constant. In higher dimensions, theexact solutions are generalizations of this wave function. In the absence ofself-interactions, the confining potential is similar to a smoothed out finitesquare well with minima also at the edges. When self-interactions are added, aterm proportional to $\pm g \psi^{\ast}\psi$ gets added to the confiningpotential and $\pm g M$, where $M$ is the norm, gets added to the total energy.In the realm of stability analysis, we find (linearly) stable solutions in thecase with repulsive self-interactions which also are stable to self-similardeformations. For attractive interactions, however, the minima at the edges ofthe potential get deeper and a barrier in the center forms as we increase thenorm. This leads to instabilities at a critical value of $M$ (related to thenumber of particles in the BEC). Comparing the stability criteria fromDerrick's theorem and Bogoliubov-de Gennes analysis stability results, we findthat both predict stability for repulsive self-interactions and instability ata critical mass $M$ for attractive interactions. However, the numericalanalysis gives a much lower critical mass. The numerical analysis shows furtherthat the initial instabilities violate the symmetry $x\rightarrow-x$ assumed byDerrick's theorem.

Uniform Bose-Einstein Condensates as Kovaton solutions of the Gross-Pitaevskii Equation through a Reverse-Engineered Potential