Three-component nonlinear Schrödinger equations: modulational instability, \(N\)th-order vector rational and semi-rational rogue waves, and dynamics. (English) Zbl 1470.35343
Summary: The integrable three-component nonlinear Schrödinger equations are systemically explored in this paper. We firstly find the conditions for the modulational instability of plane-wave solutions of the system. Secondly, we present the general formulae for the \(N\)th-order vector rational and semi-rational rogue wave solutions by the generalized Darboux transformation and formal series method. Particularly, we find that the second-order vector rational RWs contain five, seven, and nine fundamental vector RWs, which can arrange with many novel excitation dynamical patterns such as pentagon, triangle, ‘clawlike’, line, hexagon, arrow, and trapezoid structures. Moreover, we also find two different kinds of \(N\)th-order vector semi-rational RWs: one of which can demonstrate the coexistence of \(N\)th-order vector rational RW and \(N\) parallel vector breathers and the other can demonstrate the coexistence of \(N\)th-order vector rational RWs and \(N\)th-order Y-shaped vector breathers. We also exhibit distribution patterns of superposition of RWs, which can be constituted of different fundamental RW patterns. Finally, we numerically explore the dynamical behaviors of some chosen RWs. The results could excite the interest in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
three-component nonlinear Schrödinger equations; modulational instability; Darboux transformation; \(N\) th-order vector rational and semi-rational rogue waves; superposition of rogue waves; dynamicsReferences:
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