Nonlinear mirror image method for nonlinear Schrödinger equation: absorption/emission of one soliton by a boundary. (English) Zbl 1529.35463
Summary: We perform the analysis of the focusing nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of Bäcklund transformations. The difficulty arising from having such time-dependent boundary conditions at \(x = 0\) is overcome by changing the viewpoint of the method and fixing the Bäcklund transformation at infinity as well as relating its value at \(x = 0\) to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line. The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions.
{© 2021 Wiley Periodicals LLC}
{© 2021 Wiley Periodicals LLC}
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35C08 | Soliton solutions |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35A22 | Transform methods (e.g., integral transforms) applied to PDEs |
Keywords:
integrable initial-boundary value problem; inverse scattering method; nonlinear Schrödinger equationReferences:
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