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Exactly Solvable Models and Bifurcations: the Case of the CubicNLS with aδ or aδ′ Interaction in DimensionOne

Published online by Cambridge University Press:  17 July 2014

R. Adami  and
D. Noja
R. Adami
Affiliation:
Dipartimento di Scienze Matematiche “G.L. Lagrange”, Politecnico di Torino, Torino, Italy
D. Noja*
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, Italy
*
⋆Corresponding author. E-mail: riccardo.adami@polito.it, Corresponding author. E-mail: diego.noja@unimib.it
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Abstract

We explicitly give all stationary solutions to the focusing cubic NLS on the line, in thepresence of a defect of the type Dirac’s delta or delta prime. The models proveinteresting for two features: first, they are exactly solvable and all quantities can beexpressed in terms of elementary functions. Second, the associated dynamics is far frombeing trivial. In particular, the NLS with a delta prime potential shows two symmetrybreaking bifurcations: the first concerns the ground state and was already known. Thesecond emerges on the first excited state, and up to now had not been revealed. Wehighlight such bifurcations by computing the nonlinear and the no-defect limits of thestationary solutions.

Keywords

cubic NLSstationary solutionssymmetry-breaking bifurcation

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Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 1 - 16
Copyright
© EDP Sciences, 2014

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