Modulation instability and two types of non-autonomous rogue waves for the variable-coefficient AB system in fluid mechanics and nonlinear optics

2016 ◽  
Vol 30 (01) ◽  
pp. 1550264 ◽  
Author(s):  
Lei Wang ◽  
Feng-Hua Qi ◽  
Bing Tang ◽  
Yu-Ying Shi
Keyword(s):  
Nonlinear Optics ◽  
Variable Coefficient ◽  
Modulation Instability ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Variable Coefficients ◽  
Short Pulses ◽  
Plane Wave Solution ◽  
Double Peak Structure ◽  
Peak Structure

Under investigation in this paper is a variable-coefficient AB (vcAB) system, which describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. The modulation instability analysis of solutions with variable coefficients in the presence of a small perturbation is studied. The modified Darboux transformation (mDT) of the vcAB system is constructed via a gauge transformation. The first-order non-autonomous rogue wave solutions of the vcAB system are presented based on the mDT. It is found that the wave amplitude of [Formula: see text] exhibits two types of structures, i.e. the double-peak structure appears if the plane-wave solution parameter [Formula: see text] is equal to zero, while selecting [Formula: see text] yields a single-peak one. Effects of the variable coefficients on the rogue waves are graphically discussed in detail. The periodic rogue wave and composite rogue wave are obtained with different inhomogeneous parameters. Additionally, the nonlinear tunneling of the rogue waves through a conventional hyperbolic nonlinear well and barrier are investigated.

Integrable aspects and rogue wave solution of Sasa–Satsuma equation with variable coefficients in the inhomogeneous fiber

2018 ◽  
Vol 32 (05) ◽  
pp. 1850059 ◽  
Author(s):  
Yu-Ping Zhang ◽  
Lan Yu ◽  
Guang-Mei Wei
Keyword(s):  
Rogue Wave ◽  
Deep Ocean ◽  
Ocean Waves ◽  
Lax Pair ◽  
Variable Coefficients ◽  
Optical Fiber Communications ◽  
Short Pulses ◽  
Double Peak Structure ◽  
Peak Structure ◽  
Fiber Communications

Under investigation with symbolic computation in this paper, is a variable-coefficient Sasa–Satsuma equation (SSE) which can describe the ultra short pulses in optical fiber communications and propagation of deep ocean waves. By virtue of the extended Ablowitz–Kaup–Newell–Segur system, Lax pair for the model is directly constructed. Based on the obtained Lax pair, an auto-Bäcklund transformation is provided, then the explicit one-soliton solution is obtained. Meanwhile, an infinite number of conservation laws in explicit recursion forms are derived to indicate its integrability in the Liouville sense. Furthermore, exact explicit rogue wave (RW) solution is presented by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the RW can exhibit graphically an intriguing twisted rogue-wave (TRW) pair that involve four well-defined zero-amplitude points.

Lump and rogue waves for the variable-coefficient Kadomtsev–Petviashvili equation in a fluid

2018 ◽  
Vol 32 (10) ◽  
pp. 1850086 ◽  
Author(s):  
Xiao-Yue Jia ◽  
Bo Tian ◽  
Zhong Du ◽  
Yan Sun ◽  
Lei Liu
Keyword(s):  
Small Amplitude ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Single Layer ◽  
Rogue Waves ◽  
Variable Coefficients ◽  
The Other ◽  
Transverse Coordinate ◽  
Petviashvili Equation

Under investigation in this paper is the variable-coefficient Kadomtsev–Petviashvili equation, which describes the long waves with small amplitude and slow dependence on the transverse coordinate in a single-layer shallow fluid. Employing the bilinear form and symbolic computation, we obtain the lump, mixed lump-stripe soliton and mixed rogue wave-stripe soliton solutions. Discussions indicate that the variable coefficients are related to both the lump soliton’s velocity and amplitude. Mixed lump-stripe soliton solutions display two different properties, fusion and fission. Mixed rogue wave-stripe soliton solutions show that a rogue wave arises from one of the stripe solitons and disappears into the other. When the time approaches 0, rogue wave’s energy reaches the maximum. Interactions between a lump soliton and one-stripe soliton, and between a rogue wave and a pair of stripe solitons, are shown graphically.

Rogue Wave and Breather Structures with “High Frequency” and “Low Frequency” in 𝒫𝒯-Symmetric Nonlinear Couplers with Gain and Loss

2016 ◽  
Vol 71 (10) ◽  
pp. 961-969 ◽  
Author(s):  
Dang-Jun Yu ◽  
Jie-Fang Zhang
Keyword(s):  
High Frequency ◽  
Rogue Wave ◽  
Low Frequency ◽  
Zero Solution ◽  
Rogue Waves ◽  
Transformation Method ◽  
Rational Solutions ◽  
Plane Wave Solution ◽  
Gain And Loss ◽  
Basic Characteristics

AbstractBased on the modified Darboux transformation method, starting from zero solution and the plane wave solution, the hierarchies of rational solutions and breather solutions with “high frequency” and “low frequency” of the coupled nonlinear Schrödinger equation in parity-time symmetric nonlinear couplers with gain and loss are constructed, respectively. From these results, some basic characteristics of multi-rogue waves and multi-breathers are studied. Based on the property of rogue wave as the “quantum” of pattern structure in rogue wave hierarchy, we further study the novel structures of the superposed Akhmediev breathers, Kuznetsov-Ma solitons and their combined structures. It is expected that these results may give new insight into the context of the optical communications and Bose-Einstein condensations.

A variable coefficient nonlinear Schrödinger equation: localized waves on the plane wave background and their dynamics

2019 ◽  
Vol 33 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Variable Coefficient ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Plane Wave Background ◽  
Localized Waves ◽  
Similarity Reductions

In this work, a variable coefficient nonlinear Schrödinger (vc-NLS) equation is under investigation, which can describe the amplification or absorption of pulses propagating in an optical fiber with distributed dispersion and nonlinearity. By means of similarity reductions, a similar transformation helps us to relate certain class of solutions of the standard NLS equation to the solutions of integrable vc-NLS equation. Furthermore, we analytically consider nonautonomous breather wave, rogue wave solutions and their interactions in the vc-NLS equation, which possess complicated wave propagation in time and differ from the usual breather waves and rogue waves. Finally, the main characteristics of the rational solutions are graphically discussed. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

Breathers, rogue waves and their dynamics in a (2+1)-dimensional nonlinear Schrödinger equation

2020 ◽  
Vol 34 (23) ◽  
pp. 2050234
Author(s):  
Yong Chen ◽  
Xiu-Bin Wang ◽  
Bo Han
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Nonlinear Schrodinger Equation ◽  
Rational Solutions ◽  
Plane Wave Solution ◽  
Nonlinear Schrödinger ◽  
Wave Phenomena

Under investigation in this paper is a (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger equation, which is a generalization of the standard nonlinear Schrödinger equation. By means of the modified Darboux transformation, the hierarchies of rational solutions and breather solutions are generated from the plane wave solution. Furthermore, the main characteristics of the nonlinear waves including the Akhmediev breathers, Kuznetsov–Ma solitons, and their combined structures are graphically discussed. Our results would be of much importance in enriching and explaining rogue wave phenomena in nonlinear wave fields.

Rogue waves in Lugiato-Lefever equation with variable coefficients

Open Physics ◽  
2014 ◽  
Vol 12 (11) ◽  
Author(s):  
Guy Kol ◽  
Sifeu Kingni ◽  
Paul Woafo
Keyword(s):  
Pump Power ◽  
Continuous Wave ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Variable Coefficients ◽  
Constructive Method ◽  
Explicit Solutions ◽  
Continuous Wave Laser ◽  
Wave Solutions ◽  
Wave Laser

AbstractIn this paper, we theoretically investigate the generation of optical rogue waves from a Lugiato-Lefever equation with variable coefficients by using the nonlinear Schrödinger equation-based constructive method. Exact explicit rogue-wave solutions of the Lugiato-Lefever equation with constant dispersion, detuning and dissipation are derived and presented. The bright rogue wave, intermediate rogue wave and the dark rogue wave are obtained by changing the value of one parameter in the exact explicit solutions corresponding to the external pump power of a continuous-wave laser.

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