Rogue-Wave Solutions for an Inhomogeneous Nonlinear System in a Geophysical ﬂuid Or Inhomogeneous Optical Medium

Commun Nonlinear Sci Numer Simulat 36 (2016) 266–272

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier.com/locate/cnsns

Rogue-wave solutions for an inhomogeneous nonlinear system in a geophysical ﬂuid or inhomogeneous optical medium

Xi-Yang Xie a, Bo Tian a,∗, Yan Jiang a, Wen-Rong Sun a, Ya Sun a, Yi-Tian Gao b a State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China b Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China article info abstract

Article history: Under investigation in this paper is an inhomogeneous nonlinear system, which describes Received 23 September 2015 the marginally-unstable baroclinic wave packets in a geophysical fluid or ultra-short pulses Accepted 3 December 2015 in nonlinear optics with certain inhomogeneous medium existing. By virtue of a kind of the Availableonline12December2015 Darboux transformation, under the Painlevé integrable condition, the first- and second-order bright and dark rogue-wave solutions are derived. Properties of the first- and second-order Keywords: α β Inhomogeneous nonlinear system bright and dark rogue waves with (t), which measures the state of the basic flow, and (t), Baroclinic wave packets in geophysical fluids representing the interaction of the wave packet and mean flow, are graphically presented and Ultra-short pulses in nonlinear optics analyzed: α(t)andβ(t) have no influence on the wave packet, but affect the correction of the Rogue-wave solution basic flow. When we choose α(t) as a constant and linear function, respectively, the shapes Darboux transformation of the first- and second-order dark rogue waves change, and the peak heights and widths of them alter with the value of β(t) changing. © 2015 Elsevier B.V. All rights reserved.

1. Introduction

Oceanic rogue waves are the isolated waves with amplitudes much larger than the average wave crests [1–3]. Attention has been paid to the rogue waves [4], such as the observation of optical rogue waves in a photonic crystal ﬁber [5],resurgenceof surface rogue waves in a water wave tank [6], and appearance of spatiotemporal rogue waves via optical ﬁlamentation [7]. The nonlinear system [8–12],

Axt = δA + ξ AB, 1 B =− χ(|A|2 ) , (1) x 2 t describes the temporal and spatial evolution of a marginally-unstable baroclinic wave packet in a geophysical fluid or ultra-short pulse in nonlinear optics, where x and t are the normalized space and retarded time coordinates, respectively, δ is a parameter measuring the state of the basic flow, which is introduced to ideally characterize the macroscopic flow of the atmosphere, ξ is a parameter that reflects the interaction between the wave packet and mean flow, χ represents the group velocity, A is the amplitude of the wave packet while B is a quantity measuring the correction of the basic flow, and they obey the normalization condition: 2 2 |At | + B = 1. (2)

∗ Corresponding author. E-mail address: [email protected] (B. Tian). http://dx.doi.org/10.1016/j.cnsns.2015.12.004 1007-5704/© 2015 Elsevier B.V. All rights reserved. X.-Y. Xie et al. / Commun Nonlinear Sci Numer Simulat 36 (2016) 266–272 271

Fig. 3. The second-order rogue waves via Solutions (24) with a = 1, b(t) = 2t, α(t) = 1andβ(t) = 1.

Fig. 4. The same as Fig. 3(b) except that, respectively, (a) α(t) = 0.01t;(b)β(t) = 1.5.

5. Conclusions

In this paper, we have investigated System (3), an inhomogeneous nonlinear system, which describes the marginally-unstable baroclinic wave packets in a geophysical fluid or ultra-short pulses in nonlinear optics with certain inhomogeneous medium existing. Under Condition (4),viaDTs(12) and (16), the first- and second-order bright and dark rogue-wave solutions, i.e., Solutions (23) and (24), have been obtained. Properties of the first- and second-order bright and dark rogue waves have been graphically presented and analyzed with α(t), which measures the state of the basic flow, and β(t), representing the interaction of the wave packet and mean flow. α(t)andβ(t) have no influence on the wave packet, A,butaffectB, a quantity measuring the correction of the basic flow. Figs. 1 and 2 have shown the first-order bright and dark rogue waves. Fig. 1(a) has displayed the bright rogue wave with two troughs with α(t)andβ(t) as the constants. Figs. 1(b) and 2 have shown the effects of α(t)andβ(t)ontheshapesof the first-order dark rogue waves. Compared with Fig. 1(b), the shape of the dark rogue wave in Fig. 2(a) has altered when α(t)is chosen differently, and we have also observed that β(t) influences the peak height and width of the first-order dark rogue wave, comparing Fig. 2(b) with Fig. 1(b). Fig. 3(a) has shown the shapes of the second-order bright rogue waves, while the second-order dark rogue waves have been displayed in Fig. 3(b) with α(t)andβ(t) both as the constants. Compared with those in Fig. 3(b), when we choose α(t) as a linear function of t, the shapes of the dark rogue waves in Fig. 4(a) have changed. In Fig. 4(b), we have found that the peak heights and widths of the dark rogue waves change when β(t)ischosenfrom1to1.5.

Acknowledgments

This work has been supported by the National Natural Science Foundation of China under Grant nos. 11272023 and 11471050, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under Grant no. IPOC2013B008, and by the Fundamental Research Funds for the Central Universities of China under Grant nos. 2011BUPTYB02 and 2015RC19.

References

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