Modulation instability, bifurcation analysis and solitonic waves in nonlinear optical media with odd-order dispersion. (English) Zbl 1535.81109
Summary: In this paper, modulation instability, bifurcation analysis, and soliton solutions are investigated in nonlinear media with odd-order dispersion terms. A generalized nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity is considered. A linear analysis method is used to derive an expression of the modulation instability spectrum, and the effects of the fourth-order and group velocity dispersion are pointed out on the modulation instability bands. The results show that for negative values of the fourth order in a normal dispersion regime, the modulation instability vanishes. Another important aspect is revealed when the fourth-order dispersion gets positive and has high values; additional bands emerge to enlarge the bandwidths of the modulation instability. To further confirm the effects of the odd-order dispersion in the nonlinear structure, the bifurcation, phase portraits, and chaotic behaviors have been investigated to show how instability arises in the nonlinear structure. Using numerical simulation, in particular the Runge-Kutta algorithm, the sensitivity of nonlinear systems is pointed out to confirm an unstable behavior. This investigation confirms once again the fact that the modulation spectrum is sensitive to higher-order dispersion in normal and anomalous dispersion regimes. A traveling wave hypothesis is employed to lead to the direct integration of the nonlinear system, and some specific soliton solutions are extracted. For particular constraint conditions on the discriminant, bright and dark solitons as well as Jacobi elliptic function solutions emerge. The obtained results could be used to improve the transmission signal via the optical fiber and secure the data in communication systems.
MSC:
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 94A14 | Modulation and demodulation in information and communication theory |
| 70K50 | Bifurcations and instability for nonlinear problems in mechanics |
| 37G10 | Bifurcations of singular points in dynamical systems |
| 81Q50 | Quantum chaos |
| 35C08 | Soliton solutions |
| 78A60 | Lasers, masers, optical bistability, nonlinear optics |
| 78A48 | Composite media; random media in optics and electromagnetic theory |
| 81U30 | Dispersion theory, dispersion relations arising in quantum theory |
| 76F65 | Direct numerical and large eddy simulation of turbulence |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
modulation instability; bifurcation and chaotic behaviors; soliton waves; nonlinear optical mediaReferences:
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