Vector NLS solitons interacting with a boundary. (English) Zbl 1521.35168
Summary: We construct multi-soliton solutions of the \(n\)-component vector nonlinear Schrödinger equation on the half-line subject to two classes of integrable boundary conditions (BCs): the homogeneous Robin BCs and the mixed Neumann/Dirichlet BCs. The construction is based on the so-called dressing the boundary, which generates soliton solutions by preserving the integrable BCs at each step of the Darboux-dressing process. Under the Robin BCs, examples, including boundary-bound solitons, are explicitly derived; under the mixed Neumann/Dirichlet BCs, the boundary can act as a polarizer that tunes different components of the vector solitons. Connection of our construction to the inverse scattering transform is also provided.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 35Q51 | Soliton equations |
Keywords:
polarizer effect; solitons on the half-line; vector nonlinear Schrödinger equation; integrable boundary conditions; boundary-bound statesReferences:
| [1] | Sklyanin, E K1987Boundary conditions for integrable equationsFunct. Anal. Appl.21164166164-6; Sklyanin1988Boundary conditions for integrable quantum systemsJ. Phys. A: Math. Gen.212375 · Zbl 0643.35093 |
| [2] | Cherednik, I. V., Factorizing particles on a half-line and root systems, Theor. Math. Phys., 61, 977-983 (1984) · Zbl 0685.58058 · doi:10.1007/BF01038545 |
| [3] | Avan, J.; Caudrelier, V.; Crampé, N., From Hamiltonian to zero curvature formulation for classical integrable boundary conditions, J. Phys. A: Math. Theor., 51, 30LT01 (2018) · Zbl 0575.22021 · doi:10.1088/1751-8121/aac976 |
| [4] | Manakov, S. V., On the theory of two-dimensional stationary self-focusing electro-magnetic waves, Sov. Phys.-JETP, 38, 248-253 (1974) · Zbl 1407.37091 |
| [5] | Stegeman, G. I.; Segev, M., Optical spatial solitons and their interactions: universality and diversity, Science, 286, 1518-1523 (1999) · doi:10.1126/science.286.5444.1518 |
| [6] | Chen, Z.; Segev, M., Optical spatial solitons: historical overview and recent advances, Rep. Prog. Phys., 75 (2012) · doi:10.1088/0034-4885/75/8/086401 |
| [7] | Veselov, A. P., Yang-Baxter maps and integrable dynamics, Phys. Lett. A., 314, 214-221 (2003) · doi:10.1016/S0375-9601(03)00915-0 |
| [8] | Tsuchida, T., N-soliton collision in the Manakov model, Prog. Theor. Phys., 111, 151 (2004) · Zbl 1051.81014 · doi:10.1143/PTP.111.151 |
| [9] | Ablowitz, M. J.; Prinari, B.; Trubatch, A. D., Soliton interactions in the vector NLS equation, Inverse Prob., 20, 1217 (2004) · Zbl 1051.35092 · doi:10.1088/0266-5611/20/4/012 |
| [10] | Caudrelier, V.; Zhang, C., Yang-Baxter and reflection maps from vector solitons with a boundary, Nonlinearity, 27, 1081 (2014) · Zbl 1074.35082 · doi:10.1088/0951-7715/27/6/1081 |
| [11] | Caudrelier, V.; Zhang, C., Vector nonlinear Schrödinger equation on the half-line, J. Phys. A: Math Theor., 45 (2012) · Zbl 1291.35259 · doi:10.1088/1751-8113/45/10/105201 |
| [12] | Biondini, G.; Hwang, G., Boundary value problems and a nonlinear method of images, J. Phys A: Math. Theor., 42, 205-207 (2009) · Zbl 1238.35145 · doi:10.1088/1751-8113/42/20/205207 |
| [13] | Bikbaev, R. F.; Tarasov, V. O., Initial boundary value problem for the nonlinear Schrödinger equation, J. Phys. A: Math. Gen., 24, 2507 (1991) · Zbl 1173.35675 · doi:10.1088/0305-4470/24/11/017 |
| [14] | Zhang, C., Dressing the boundary: on soliton solutions of the nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 142, 190-212 (2019) · Zbl 0753.35090 · doi:10.1111/sapm.12248 |
| [15] | Zhang, C.; Cheng, Q.; Zhang, D. J., Soliton solutions of the sine-Gordon equation on the half line, Appl. Math. Lett., 86, 64-69 (2018) · Zbl 1420.35388 · doi:10.1016/j.aml.2018.06.020 |
| [16] | Fokas, A. S., A unified transform method for solving linear and certain nonlinear PDEs, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci., 453, 1411-1443 (1997) · Zbl 1407.35052 · doi:10.1098/rspa.1997.0077 |
| [17] | Fokas, A. S., Integrable nonlinear evolution equations on the half-line, Commun. Math. Phys., 230, 1-39 (2002) · Zbl 0876.35102 · doi:10.1007/s00220-002-0681-8 |
| [18] | Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, 1095 (1967) · Zbl 1010.35089 · doi:10.1103/PhysRevLett.19.1095 |
| [19] | Zakharov, V.; Shabat, A., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys., 34, 62-69 (1972) · Zbl 1061.35520 |
| [20] | Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53, 249-315 (1974) · doi:10.1002/sapm1974534249 |
| [21] | Faddeev, L.; Takhtajan, L., Hamiltonian Methods in the Theory of Solitons (2007), Berlin: Springer, Berlin · Zbl 0408.35068 · doi:10.1007/978-3-540-69969-9 |
| [22] | Fokas, A. S.; Its, A. R.; Sung, L. Y., The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18, 1771 (2005) · Zbl 1111.37001 · doi:10.1088/0951-7715/18/4/019 |
| [23] | Geng, X.; Liu, H.; Zhu, J., Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line, Stud. Appl. Math., 135, 310-346 (2015) · Zbl 1181.37095 · doi:10.1111/sapm.12088 |
| [24] | Zakharov, VShabat, A1974Scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. IFunct. Anal. Appl.8226235226-35; Zakharov, S1979Integration of nonlinear equations of mathematical physics by the method of inverse scattering. IIFunct. Anal. Appl.13166174166-74 · Zbl 1338.35408 |
| [25] | Matveev, V. B.; Salle, M. A., Darboux Transformations and Solitons (1991), Berlin: Springer, Berlin · Zbl 0303.35024 |
| [26] | Babelon, O.; Bernard, D.; Talon, M., Introduction to Classical Integrable Systems (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 0448.35090 · doi:10.1017/CBO9780511535024 |
| [27] | Cieśliński, J. L., Algebraic construction of the Darboux matrix revisited, J. Phys. A: Math. Theor., 42 (2009) · Zbl 0744.35045 · doi:10.1088/1751-8113/42/40/404003 |
| [28] | Habibullin, I. T.; Svinolupov, S., Integrable boundary value problems for the multicomponent Schrödinger equations, Physica D., 87, 134-139 (1995) · Zbl 1045.37033 · doi:10.1016/0167-2789(95)00165-Z |
| [29] | Biondini, G.; Bui, A., On the nonlinear Schrödinger equation on the half line with homogeneous Robin boundary conditions, Stud. Appl. Math., 129, 249-271 (2012) · Zbl 1193.37096 · doi:10.1111/j.1467-9590.2012.00553.x |
| [30] | Deift, P.; Park, J., Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data, Int. Math. Res. Notices., 2011, 5505-5624 (2011) · Zbl 1194.35411 · doi:10.1093/imrn/rnq282 |
| [31] | Biondini G and Bui A 2012 On the nonlinear Schrödinger equation on the half line with homogeneous Robin boundary conditions Stud. Appl. Math.129 249-71 · Zbl 1277.35312 · doi:10.1111/j.1467-9590.2012.00553.x |
| [32] | Deift P and Park J 2011 Long-time asymptotics for solutions of the NLS equation with a delta potential and even initial data Int. Math. Res. Notices.2011 5505-624 · Zbl 1251.35145 · doi:10.1093/imrn/rnq282 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
