Integrability, modulational instability and mixed localized wave solutions for the generalized nonlinear Schrödinger equation. (English) Zbl 1491.35394
Summary: Under investigation in this paper is the generalized nonlinear Schrödinger (g-NLS) equation which has extensive applications in various physical fields. Firstly, we prove Liouville integrability of this equation by deriving its bi-Hamiltonian structures applying the variational identity. Nextly, we calculate the modulational instability for the possible reason of the formation of the rogue wave. Moreover, based on the generalized \((2, N-2)\)-fold Darboux transformation (DT), we can derive several mixed localized wave solutions such as breathers, rogue waves and semi-rational solitons for this equation, and accurately analyze a lot of important physical quantities. Finally, we present these solutions graphically by choosing appropriate parameters and discuss their dynamic behavior. It is worth noting that all of these solutions can change from a strong interaction to a weak interaction by choosing the parameters. This may also be one of the reasons why relevant wave structures presenting diversity, and useful to explain some physical phenomena in nonlinear optics.
Editorial remark. See the comment by E. Kengne on this paper [Z. Angew. Math. Phys. 75, No. 4, Paper No. 138, 8 p. (2024; Zbl 07900954)].
Editorial remark. See the comment by E. Kengne on this paper [Z. Angew. Math. Phys. 75, No. 4, Paper No. 138, 8 p. (2024; Zbl 07900954)].
MSC:
| 35Q51 | Soliton equations |
| 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.) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
| 35B35 | Stability in context of PDEs |
| 35C08 | Soliton solutions |
| 41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
Keywords:
generalized nonlinear Schrödinger equation; Liouville integrability; modulational instability; generalized \((m; N-m)\)-fold DT; mixed localized wave solutionsCitations:
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