Discrete Derivative Nonlinear Schrödinger Equations
We consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central difference schemes, respectively, and term the corresponding equat...
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MDPI AG
2024-12-01
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| author | Dirk Hennig Jesús Cuevas-Maraver |
| author_facet | Dirk Hennig Jesús Cuevas-Maraver |
| author_sort | Dirk Hennig |
| collection | DOAJ |
| description | We consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central difference schemes, respectively, and term the corresponding equations forward, backward, and central ddNLSs. We show that in contrast to the dNLS, which is completely integrable and supports soliton solutions, the forward and backward ddNLSs can be either dissipative or expansive. As a consequence, solutions of the forward and backward ddNLSs behave drastically differently compared to those of the (integrable) dNLS. For the dissipative forward ddNLS, all solutions decay asymptotically to zero, whereas for the expansive forward ddNLS all solutions grow exponentially in time, features that are not present in the dynamics of the (integrable) dNLS. In comparison, the central ddNLS is characterized by conservative dynamics. Remarkably, for the central ddNLS the total momentum is conserved, allowing the existence of solitary travelling wave (TW) solutions. In fact, we prove the existence of solitary TWs, facilitating Schauder’s fixed-point theorem. For the damped forward expansive ddNLS we demonstrate that there exists such a balance of dissipation so that solitary stationary modes exist. |
| format | Article |
| id | doaj-art-460f06ec085e4f818ca9d58bf40173ad |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-460f06ec085e4f818ca9d58bf40173ad2025-01-10T13:18:16ZengMDPI AGMathematics2227-73902024-12-0113110510.3390/math13010105Discrete Derivative Nonlinear Schrödinger EquationsDirk Hennig0Jesús Cuevas-Maraver1Department of Mathematics, University of Thessaly, 35100 Lamia, GreeceGrupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/Virgen de África, 7, 41011 Sevilla, SpainWe consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central difference schemes, respectively, and term the corresponding equations forward, backward, and central ddNLSs. We show that in contrast to the dNLS, which is completely integrable and supports soliton solutions, the forward and backward ddNLSs can be either dissipative or expansive. As a consequence, solutions of the forward and backward ddNLSs behave drastically differently compared to those of the (integrable) dNLS. For the dissipative forward ddNLS, all solutions decay asymptotically to zero, whereas for the expansive forward ddNLS all solutions grow exponentially in time, features that are not present in the dynamics of the (integrable) dNLS. In comparison, the central ddNLS is characterized by conservative dynamics. Remarkably, for the central ddNLS the total momentum is conserved, allowing the existence of solitary travelling wave (TW) solutions. In fact, we prove the existence of solitary TWs, facilitating Schauder’s fixed-point theorem. For the damped forward expansive ddNLS we demonstrate that there exists such a balance of dissipation so that solitary stationary modes exist.https://www.mdpi.com/2227-7390/13/1/105discrete derivative nonlinear Schrödinger equationssolitonsasymptotic behaviour of solutionstravelling solitary waves |
| spellingShingle | Dirk Hennig Jesús Cuevas-Maraver Discrete Derivative Nonlinear Schrödinger Equations Mathematics discrete derivative nonlinear Schrödinger equations solitons asymptotic behaviour of solutions travelling solitary waves |
| title | Discrete Derivative Nonlinear Schrödinger Equations |
| title_full | Discrete Derivative Nonlinear Schrödinger Equations |
| title_fullStr | Discrete Derivative Nonlinear Schrödinger Equations |
| title_full_unstemmed | Discrete Derivative Nonlinear Schrödinger Equations |
| title_short | Discrete Derivative Nonlinear Schrödinger Equations |
| title_sort | discrete derivative nonlinear schrodinger equations |
| topic | discrete derivative nonlinear Schrödinger equations solitons asymptotic behaviour of solutions travelling solitary waves |
| url | https://www.mdpi.com/2227-7390/13/1/105 |
| work_keys_str_mv | AT dirkhennig discretederivativenonlinearschrodingerequations AT jesuscuevasmaraver discretederivativenonlinearschrodingerequations |