Publisher: Institute of Physics
Languages: English
Types: Article
Subjects: QA, QC, Mathematical Physics, Nonlinear Sciences  Exactly Solvable and Integrable Systems, High Energy Physics  Theory
Identifiers:doi:10.1088/17518113/45/10/105201
We investigate the Manakov model or, more generally, the vector nonlinear Schr\"odinger equation on the halfline. Using a B\"acklund transformation method, two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet and Robin boundary conditions. Integrability is shown by constructing a generating function for the conserved quantities. We apply a nonlinear mirror image technique to construct the inverse scattering method with these boundary conditions. The important feature in the reconstruction formula for the fields is the symmetry property of the scattering data emerging from the presence of the boundary. Particular attention is paid to the discrete spectrum. An interesting phenomenon of transmission between the components of a vector soliton interacting with the boundary is demonstrated. This is specific to the vector nature of the model and is absent in the scalar case. For onesoliton solutions, we show that the boundary can be used to make certain components of the incoming soliton vanishingly small. This is reminiscent of the phenomenon of light polarization by reflection.

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 [1] S. Manakov, On the theory of twodimensional stationary selffocusing of electromagnetic waves, Sov. Phys. JETP 38, (1974), 248.
 [2] E. H. Lieb, W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the ground state, Phys. Rev. 130 No. 4 (1963) 1605.
 [3] C. N. Yang, Some exact results for the manybody problem in one dimension with repulsive deltafunction interaction, Phys. Rev. Lett. 19 (1967), 1312.
 [4] E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A21 (1988), 2375.
 [5] S. Murakami, M. Wadati, Connection between Yangian symmetry and the quantum inverse scattering method, J. Phys. A29 (1996), 7903.
 [6] M. Mintchev, É. Ragoucy, P. Sorba, Ph. Zaugg, Yangian symmetry in the Non Linear Schrödinger hierarchy, J. Phys. A32 (1999), 5885.
 [7] M. Mintchev, É. Ragoucy, P. Sorba, Spontaneous symmetry breaking in the gl(N) NLS hierarchy on the half line, J. Phys. A 34 (2001), 8345.
 [8] A. P. Fordy, P. P. Kulish,Nonlinear Schrödinger equations and simple Lie algebras, Comm. Math. Phys. 89, (1983), 427.
 [9] T. Tsuchida, Nsoliton collision in the Manakov model, Prog. Theor. Phys. 111, (2004), 151.
 [10] I. T. Habibullin, S. I. Svinolupov, Integrable boundary value problems for the multicomponent Schrödinger equations, Physica D 87, (1995), 134.
 [11] M. Ablowitz, H. Segur, The inverse scattering transform: Semiinfinite interval , J. Math. Phys. 16, (1975), 1054.
 [12] A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMSSIAM (2008).
 [13] I. T. Habibullin, Bäcklund transformation and integrable boundaryinitial value problems, in Nonlinear world", vol. 1 (Kiev, 1989), 130138, World Sci. Publ., River Edge, NJ, 1990; Integrable initialboundary value problems, Theor. and Math. Phys. 86 (1991),28.
 [14] P. N. Bibikov, V. O. Tarasov, A boundaryvalue problem for the nonlinear Schrödinger equation, Theor. Math. Phys. 79 (1989), 334.
 [15] R. F. Bikbaev, V. O. Tarasov, Initialboundary value problem for the nonlinear Schrödinger equation, J. Phys. A24 (1991) 2507.
 [16] A. S. Fokas, An initialboundary value problem for the nonlinear Schödinger equation, Physica D 35, (1989), 167.
 [17] G. Biondini, G. Hwang, Solitons, boundary value problems and a nonlinear method of images, J. Phys. A42, (2009), 205207.
 [18] A. S. Fokas, A. Its, The linearization of the initialboundary value problem of the nonlinear Schrödinger equation, SIAM J. Math. Anal. 27, (1996), 738.
 [19] M. J. Ablowitz, B. Prinari, A. D. Trubatch, Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Notes Series 302, Cambridge University Press, 2003.
 [20] E. K. Sklyanin, Canonicity of Bäcklund transformation: rmatrix approach. II, solvint/9903017.
 [21] V. Caudrelier, On a systematic approach to defects in classical integrable field theories, IJGMMP 5, (2008), 1085.

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