Opuscula Math. 38, no. 6 (2018), 883-898

Opuscula Mathematica

Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method

Yan Rybalko

Abstract. We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem.

Keywords: the Fokas unified transform method, Schrödinger equation, interface problems.

Mathematics Subject Classification: 35Q41, 35E15.

Full text (pdf)

  1. M.J. Ablowitz, H. Segur, Solitons and Inverse Scattering Transform, SIAM Studies in Applied Mathematics 4. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1981.
  2. A. Boutet de Monvel, A.S. Fokas, D. Shepelsky, Analysis of the global relation for the nonlinear Schrödinger equation on the half-line, Lett. Math. Phys. 65 (2003) 3, 199-212.
  3. A. Boutet de Monvel, A.S. Fokas, D. Shepelsky, Integrable nonlinear evolution equations on a finite interval, Commun. Math. Phys. 263 (2006), 133-172.
  4. B. Deconinck, N. Sheils, Interface problems for dispersive equations, Studies in Applied Mathematics 134 (2015) 3, 253-275.
  5. B. Deconinck, N. Sheils, Initial-to-interface maps for the heat equation on composite domains, Studies in Applied Mathematics 137 (2016) 1, 140-154.
  6. B. Deconinck, B. Pelloni, N. Sheils, Non-steady-state heat conduction in composite walls, Proceedings of the Royal Society A 470 (2014) 2165.
  7. B. Deconinck, N. Sheils, D. Smith, The linear KdV equation with an interface, Communications in Mathematical Physics 347 (2016) 2, 489-509.
  8. B. Deconinck, T. Trogdon, V. Vasan, The method of Fokas for solving linear partial differential equations, SIAM Review 56 (2014) 1, 159-186.
  9. P. Deift, J. Park, Long-time asymptotics for solutions of the NLS equation, International Mathematics Research Notices 2011 (2011) 24, 5505-5624.
  10. P.A. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Annals of Mathematics 137 (1993) 2, 295-368.
  11. P.A. Deift, A.R. Its, X. Zhou, Long-time asymptotics for integrable nonlinear wave equations, [in:] A.S. Fokas, V.E. Zakharov (eds.), Important Developments in Soliton Theory, Springer, 1993, pp. 181-204.
  12. A.S. Fokas, A unified transform method for solving linear and certain nonlinear PDE's, Proceedings of the Royal Society A 453 (1997), 1411-1443.
  13. A.S. Fokas, Integrable nonlinear evolution equations on the half-line, Communications in Mathematical Physics 230 (2002), 1-39.
  14. A.S. Fokas, A Unified Approach to Boundary Value Problems, SIAM, Philadelphia, 2007.
  15. A. Fokas, B. Pelloni, A transform method for linear evolution PDEs on a finite interval, IMA Journal of Applied Mathematics 70 (2005) 4, 564-587.
  16. A.S. Fokas, A.R. Its, L.Y. Sung, The nonlinear Schrödinger equation on the half-line, Nonlinearity 18 (2005), 1771-1822.
  17. J. Holmer, M. Zvorski, Breathing pattern in nonlinear relaxation, Nonlinearity 22 (2009), 1259-1301.
  18. A. Its, D. Shepelsky, Initial boundary value problem for the focusing nonlinear Schrödinger equation with Robin boundary condition: half-line approach, Proceedings of the Royal Society A 469 (2013) 2149.
  19. P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-González (eds.), Emergent Nonlinear Phenomena in Bose-Einstein Condensates. Atomic, Optical, and Plasma Physics, Springer, Berlin, Heidelberg 45 (2008).
  20. P.D. Lax, Integrals of nonlinear equations of evolution and solitary Waves, Communications on Pure and Applied Mathematics 21 (1968), 467-490.
  21. J. Lenells, A.S. Fokas, The unified method: II NLS on the half-line with t-periodic boundary conditions, J. Phys. A: Math. Theor. 45 (2012) 19, 195-202.
  22. J. Lenells. A.S. Fokas, The nonlinear Schrödinger equation with t-periodic data: II. Perturbative results, Proc. Roy. Soc. A 471 (2015), 2181.
  23. B. Pelloni, Advances in the study of boundary value problems for nonlinear integrable PDEs, Nonlinearity 28 (2015) 2, R1-R38.
  24. J. Rogel-Salazar, The Gross-Pitaevskii equation and Bose-Einstein condensates, European Journal of Physics 34 (2013) 2, 247-257.
  • Yan Rybalko
  • V.N. Karazin Kharkiv National University, B. Verkin Institute for Low Temperature Physics and Engineering, Ukraine
  • Communicated by P.A. Cojuhari.
  • Received: 2018-01-29.
  • Accepted: 2018-03-15.
  • Published online: 2018-07-05.
Opuscula Mathematica - cover

Cite this article as:
Yan Rybalko, Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method, Opuscula Math. 38, no. 6 (2018), 883-898, https://doi.org/10.7494/OpMath.2018.38.6.883

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.