Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 046, 17 pages      arXiv:0903.1493      https://doi.org/10.3842/SIGMA.2010.046
Contribution to the Special Issue “Noncommutative Spaces and Fields”

The Scattering Problem for a Noncommutative Nonlinear Schrödinger Equation

Bergfinnur Durhuus a and Victor Gayral b
a) Department of Mathematics, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
b) Laboratoire de Mathématiques, Université de Reims Champagne-Ardenne, Moulin de la Housse - BP 1039 51687 Reims cedex 2, France

Received March 03, 2010, in final form May 20, 2010; Published online June 03, 2010

Abstract
We investigate scattering properties of a Moyal deformed version of the nonlinear Schrödinger equation in an even number of space dimensions. With rather weak conditions on the degree of nonlinearity, the Cauchy problem for general initial data has a unique globally defined solution, and also has solitary wave solutions if the interaction potential is suitably chosen. We demonstrate how to set up a scattering framework for equations of this type, including appropriate decay estimates of the free time evolution and the construction of wave operators defined for small scattering data in the general case and for arbitrary scattering data in the rotationally symmetric case.

Key words: noncommutative geometry; nonlinear wave equations; scattering theory; Jacobi polynomials.

pdf (289 kb)   ps (195 kb)   tex (20 kb)

References

  1. Cazenave T., Lions P.L., Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549-561.
  2. Cazenave T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, Vol. 10, Amer. Math. Soc., Providence, RI, 2003.
  3. Durhuus B., Jonsson T., Noncommutative waves have infinite propagation speed, J. High Energy Phys. 2004 (2004), no. 10, 050, 13 pages, hep-th/0408190.
  4. Durhuus B., Jonsson T., Nest R., The existence and stability of noncommutative scalar solitons, Comm. Math. Phys. 233 (2003), 49-78, hep-th/0107121.
  5. Erdélyi T., Magnus A.P., Nevai P., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614.
  6. Gayral V., Gracia-Bondía J.M., Iochum B., Schücker T., Várilly J.C., Moyal planes are spectral triples, Comm. Math. Phys. 246 (2004), 569-623, hep-th/0307241.
  7. Ginibre J., An introduction to nonlinear Schrödinger equations, in Nonlinear Waves (Sappore 1995), Editors R. Agemi, Y. Giga and T. Ozawa, GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 1997, 85-133.
  8. Ginibre J., Velo G., Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), 399-442.
  9. Gopakumar R., Minwalla S., Strominger A., Noncommutative solitons, J. High Energy Phys. 2000, (2000), no. 5, 020, 27 pages, hep-th/0003160.
  10. Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160-197.
  11. Harvey J., Kraus P., Larsen F., Exact noncommutative solitons, J. High Energy Phys. 2000, (2000), no. 12, 024, 24 pages, hep-th/0010060.
  12. Krasikov I., An upper bound on Jacobi polynomials, J. Approx. Theory 149 (2007), 116-130, math.CA/0610111.
  13. Reed M., Abstract non-linear wave equations, Lecture Notes in Mathematics, Vol. 507, Springer-Verlag, Berlin - New York, 1976.
  14. Reed M., Simon B., Methods of modern mathematical physics, Vol. II, Academic Press, New York - London, 1975.
  15. Shatah J., Strauss W.A., Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173-190.
  16. Szegö G., Orthogonal polynomials, American Mathematical Society Colloquium Publications, Vol. 23, Amer. Math. Soc., Providence, RI, 1959.
  17. Strauss W.A., Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, Vol. 73, Amer. Math. Soc., Providence, RI, 1989.
  18. Wulkenhaar R., Renormalisation of noncommutative φ44-theory to all orders, Habilitation Thesis, University of Vienna, 2003, available at http://www.math.uni-muenster.de/u/raimar/.


Previous article   Next article   Contents of Volume 6 (2010)