Exactly solvable models and bifurcations: The case of the cubic NLS with a or a interaction in dimension one

We explicitly give all stationary solutions to the focusing cubic NLS on the line, in the presence of a defect of the type Dirac's delta or delta prime. The models prove interesting for two features: first, they are exactly solvable and all quantities can be expressed in terms of elementary functions. Second, the associated dynamics is far from being trivial. In particular, the NLS with a delta prime potential shows two symmetry breaking bifurcations: the first concerns the ground state and was already known. The second emerges on the first excited state, and up to now had not been revealed. We highlight such bifurcations by computing the nonlinear and the no-defect limits of the stationary solutions. © 2014 EDP Sciences