Spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schroedinger equation with periodic boundary conditions
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2023
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915717268373504 |
|---|---|
| author | Biondini, Gino Zhang, Zechuan |
| author_facet | Biondini, Gino Zhang, Zechuan |
| contents | The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for the nonlinear Schrodinger (NLS) equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schroedinger equation with periodic boundary conditions Biondini, Gino Zhang, Zechuan Analysis of PDEs Exactly Solvable and Integrable Systems The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for the nonlinear Schrodinger (NLS) equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used. |
| title | Spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schroedinger equation with periodic boundary conditions |
| topic | Analysis of PDEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2311.18127 |