Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T05:07:05.516Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected], Corresponding author. E-mail: [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adami, R., Cacciapuoti, C., Finco, D., Noja, D.. Fast solitons on star graphs. Rev. Math. Phys, 23, 4 (2011), 409–451. CrossRefGoogle Scholar
R. Adami, D. Noja. Existence of dynamics for a 1-d NLS equation in dimension one. J. Phys. A, 42, 49, 495302 (2009), 19pp.
Adami, R., Noja, D.. Stability and symmetry-breaking bifurcation for the Ground States of a NLS with a δ′ interaction. Commun. Math. Phys., 318 (2013), 247–289. CrossRefGoogle Scholar
Adami, R., Noja, D., Visciglia, N.. Constrained energy minimization and ground states for NLS with point defects. Disc. Cont. Dyn. Syst. B, 18 (2013), no. 5, 1155–1188. CrossRefGoogle Scholar
Albeverio, S., Brzeźniak, Z., Dabrowski, L.. Fundamental solutions of the Heat and Schrödinger Equations with point interaction. J. Func. An., 130 (1995), 220–254. CrossRefGoogle Scholar
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden. Solvable Models in Quantum Mechanics. Springer-Verlag, New York, 1988.
S. Albeverio, P. Kurasov. Singular Perturbations of Differential Operators. Cambridge University Press, 2000.
Cao Xiang, D., Malomed, A.B.. Soliton defect collisions in the nonlinear Schrödinger equation. Phys. Lett. A, 206 (1995), 177–182. Google Scholar
Cheon, T., Shigehara, T.. Realizing discontinuous wave functions with renormalized short-range potentials. Phys. Lett. A, 243 (1998), 111–116. CrossRefGoogle Scholar
P. Exner, S.S. Manko. Approximations of quantum-graph vertex couplings by singularly scaled rank-one operators. Lett. Math. Phys. (2014), to appear, arXiv:1310.5856.
Exner, P., Neidhardt, H., Zagrebnov, V.A.. Potential approximations to a δ′: an inverse Klauder phenomenon with norm-resolvent convergence. Commun. Math. Phys, 224 (2001), 593–612. CrossRefGoogle Scholar
Fukuizumi, R., Jeanjean, L.. Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential. Disc. Cont. Dyn. Syst. (A), 21 (2008), 129–144. Google Scholar
Fukuizumi, R., Ohta, M., Ozawa, T.. Nonlinear Schrödinger equation with a point defect. Ann. Inst. H. Poincaré - AN, 25 (2008), 837–845. CrossRefGoogle Scholar
Fukuizumi, R., Sacchetti, A.. Bifurcation and stability for nonlinear Schrödinger equation with double well potential in the semiclassical limit. J. Stat. Phys, 145 (2011), 1546-1594. CrossRefGoogle Scholar
Golovaty, Yu.D., Hryniv, R.O.. On norm resolvent convergence of Schrödinger operators with δ′-like potentials. J. Phys. A Math. Theor., 44 (2011), 049802; Corrigendum J. Phys. A Math. Theor. 44 (2011), 049802. CrossRefGoogle Scholar
Goodman, R.H., Holmes, P.J., Weinstein, M.I.. Strong NLS soliton-defect interactions. Physica D, 192 (2004), 215–248. CrossRefGoogle Scholar
Jackson, R.K., Weinstein, M.I.. Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation. J. Stat. Phys., 116 (2004), 881–905. CrossRefGoogle Scholar
Kirr, E., Kevrekidis, P.G., Pelinovsky, D.E.. Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials. Commun. Math. Phys., 308 (2011), 795–844. CrossRefGoogle Scholar
Le Coz, S., Fukuizumi, R., Fibich, G., Ksherim, B., Sivan, Y.. Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential. Physica D, 237 (2008), no. 8, 1103–1128. CrossRefGoogle Scholar
Witthaut, D., Mossmann, S., Korsch, H.J.. Bound and resonance states of the nonlinear Schrödinger equation in simple model systems. J. Phys. A, 38 (2005), 1777-1702. CrossRefGoogle Scholar
Zakharov, V.E., Shabat, A.B.. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl., 8 (1974), 226–235. CrossRefGoogle Scholar