Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation

2016 ◽  
Vol 71 (2) ◽  
pp. 95-101 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Xiao-Li Liu ◽  
Li-Li Wen
Keyword(s):  
Rogue Wave ◽  
Dimensional Space ◽  
Rogue Waves ◽  
Space Structures ◽  
Nls Equation ◽  
Third Order ◽  
Nonlinear Schrödinger ◽  
Circular Pattern ◽  
Bright Line ◽  
Fundamental Pattern

AbstractIn this paper, a (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which is a generalisation of the NLS equation, is under investigation. The classical and generalised N-fold Darboux transformations are constructed in terms of determinant representations. With the non-vanishing background and iterated formula, a family of the analytical solutions of the (2+1)-dimensional NLS equation are systematically generated, including the bright-line solitons, breathers, and rogue waves. The interaction mechanisms between two bright-line solitons and among three bright-line solitons are both elastic. Several patterns for first-, second, and higher-order rogue wave solutions fixed at space are displayed, namely, the fundamental pattern, triangular pattern, and circular pattern. The two-dimensional space structures of first-, second-, and third-order rogue waves fixed at time are also demonstrated.

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 waves for an inhomogeneous discrete nonlinear Schrödinger equation in a lattice

2019 ◽  
Vol 33 (08) ◽  
pp. 1950090
Author(s):  
Xiao-Yu Wu ◽  
Bo Tian ◽  
Zhong Du ◽  
Xia-Xia Du
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Nonlinear Schrodinger Equation ◽  
External Potential ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Circular Pattern

Lattices are used in such fields as electricity, optics and magnetism. Under investigation in this paper is an inhomogeneous discrete nonlinear Schrödinger equation, which models the wave propagation in a lattice. Employing the Kadomtsev–Petviashvili (KP) hierarchy reduction, we obtain the rogue-wave solutions, and see that the rogue waves are affected by the coefficient of the on-site external potential. We see (1) the first-order rogue wave with one peak and two hollows; (2) the second-order rogue waves, each of which is with one peak or three humps; (3) the third-order rogue waves, each of which is with one peak or six humps, and those humps exhibit the triangular pattern, anti-triangular pattern and circular pattern. When the coefficient of the on-site external potential is a constant, the rogue waves periodically appear. When the coefficient of the on-site external potential monotonously changes, oscillations emerge on the constant background.

Study on Breather-Type Rogue Wave Based on Fourth-Order Nonlinear Schrödinger Equation

2014 ◽  
Author(s):  
Wenyue Lu ◽  
Jianmin Yang ◽  
Haining Lv ◽  
Xin Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Nonlinear Schrodinger Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
The Waves

Rogue wave is a kind of wave that possesses concentrated energy, strong nonlinear and enormous devastating. When it interacts with the deep-sea structures, the structure will suffer a serious threat, and it may even cause significant harm to the offshore staff and property. Studies on the mechanism of rogue wave are of great significance to the platform design and security. It is also one of the hot issues on the waves of hydrodynamic studies. Some breather-type solutions of NLS equation have been considered as prototypes of rogue waves in ocean. They can appear from smooth initial condition only with a certain disturb given by the exact solution of NLS. In this paper, we have numerically studied rogue waves based on fourth order nonlinear Schrödinger equation. We show that the peaks of the largest amplitude of the resulting waves can be described in terms of the Peregrine breather-type solution as the solution of NLS equation.

Rogue-Wave Interaction of a Nonlinear Schrödinger Model for the Alpha Helical Protein

2016 ◽  
Vol 71 (1) ◽  
pp. 27-32 ◽  
Author(s):  
Hui-Xian Jia ◽  
Yu-Jun Liu ◽  
Ya-Ning Wang
Keyword(s):  
Rogue Wave ◽  
Wave Interaction ◽  
Rogue Waves ◽  
Higher Order ◽  
Order Effects ◽  
Third Order ◽  
Nonlinear Schrödinger ◽  
Existence Time ◽  
Helical Protein ◽  
Higher Order Effects

AbstractIn this article, we investigate a fourth-order nonlinear Schrödinger equation, which governs the Davydov solitons in the alpha helical protein with higher-order effects. By virtue of the generalised Darboux transformation, higher-order rogue-wave solutions are derived. Propagation and interaction of the rogue waves are analysed: (i) Coefficients affect the existence time of the first-order rogue waves; (ii) coefficients affect the interaction time of the second- and third-order rogue waves; (iii) direction of the rogue-wave propagation remain unchanged after interaction.

Hybrid soliton solutions in the (2+1)-dimensional nonlinear Schrödinger equation

2017 ◽  
Vol 31 (32) ◽  
pp. 1750298 ◽  
Author(s):  
Meidan Chen ◽  
Biao Li
Keyword(s):  
Rogue Wave ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Nls Equation ◽  
Bilinear Method ◽  
Nonlinear Schrödinger ◽  
Long Wave ◽  
Hybrid Solutions ◽  
Wave Limit

Rational solutions and hybrid solutions from N-solitons are obtained by using the bilinear method and a long wave limit method. Line rogue waves and lumps in the (2[Formula: see text]+[Formula: see text]1)-dimensional nonlinear Schrödinger (NLS) equation are derived from two-solitons. Then from three-solitons, hybrid solutions between kink soliton with breathers, periodic line waves and lumps are derived. Interestingly, after the collision, the breathers are kept invariant, but the amplitudes of the periodic line waves and lumps change greatly. For the four-solitons, the solutions describe as breathers with breathers, line rogue waves or lumps. After the collision, breathers and lumps are kept invariant, but the line rogue wave has a great change.

Abundant rogue wave solutions for the (2 + 1)-dimensional generalized Korteweg–de Vries equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huanhuan Lu ◽  
Yufeng Zhang
Keyword(s):  
Vries Equation ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Rogue Wave Solutions ◽  
Bilinear Formulation ◽  
Hirota Bilinear

AbstractIn this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

New Rational Homoclinic Solution and Rogue Wave Solution for the Coupled Nonlinear Schrödinger Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 441-445 ◽  
Author(s):  
Long-Xing Li ◽  
Jun Liu ◽  
Zheng-De Dai ◽  
Ren-Lang Liu
Keyword(s):  
Schrödinger Equation ◽  
Wave Solution ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Homoclinic Solution ◽  
Nonlinear Schrodinger Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Rogue Wave Solution

In this work, the rational homoclinic solution (rogue wave solution) can be obtained via the classical homoclinic solution for the nonlinear Schrödinger (NLS) equation and the coupled nonlinear Schrödinger (CNLS) equation, respectively. This is a new way for generating rogue wave comparing with direct constructing method and Darboux dressing technique

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.

scholarly journals Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

scholarly journals Multi-rogue Wave Solutions for a Generalized Integrable Discrete Nonlinear Schrodinger Equation With Higher-order Excitations

2021 ◽  
Author(s):  
Ma Li-Yuan ◽  
Yang Jun ◽  
Zhang Yan-Li
Keyword(s):  
Modulation Instability ◽  
Rogue Wave ◽  
Order Term ◽  
Higher Order ◽  
Discrete Version ◽  
Nls Equation ◽  
Third Order ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Dynamical Behaviors ◽  
Continuous Waves

Abstract In this paper, we construct the discrete rogue wave(RW) solutions for a higher-order or generalized integrable discrete nonlinear Schr¨odinger(NLS) equation. First, based on the modified Lax pair, the discrete version of generalized Darboux transformation are constructed. Second, the dynamical behaviors of first-, second- and third-order RWsolutions are investigated in corresponding to the unique spectral parameter λ, higher-order term coefficient γ, and free constants dk, fk (k = 1, 2, · · · ,N), which exhibit affluent wave structures. The differences between the RW solution of the higher-order discrete NLS equation and that of the Ablowitz-Ladik(AL) equation are illustrated in figures. Moreover, numerical experiments are explored, which demonstrates that strong-interaction RWs are stabler than the weak-interaction RWs. Finally, the modulation instability of continuous waves is studied.

Sign in / Sign up

Export Citation Format

Share Document