Novel localized wave interaction phenomena and dynamics in the generalized discrete Hirota equation via the generalized (2,N − 2)-fold Darboux transformation

2019 ◽  
Vol 33 (17) ◽  
pp. 1950192 ◽  
Author(s):  
Hao-Tian Wang ◽  
Xiao-Yong Wen ◽  
Yaqing Liu
Keyword(s):  
Darboux Transformation ◽  
Wave Interaction ◽  
Inelastic Interaction ◽  
Discrete Version ◽  
The Novel ◽  
Spectral Parameters ◽  
Small Noise ◽  
Hirota Equation ◽  
Interaction Solutions ◽  
Dynamical Behaviors

We investigate the generalized discrete Hirota equation, which is an integrable discrete version of Hirota equation. The generalized [Formula: see text]-fold Darboux transformation (i.e. generalized [Formula: see text]-fold Darboux transformation when [Formula: see text] with two spectral parameters) is presented to derive the novel localized wave interaction solutions. Moreover, some inelastic interaction evolution phenomena of rogue wave and breather are studied graphically. Numerical simulations are used to explore the dynamical behaviors of certain localized wave interaction solutions, which show that their evolutions are stable against a small noise. The results obtained in this paper can be used to find the interaction properties of localized nonlinear waves in nonlinear optics and relevant fields.

scholarly journals Nonlinear Dynamics of a Vibratory Cone Crusher with Hysteretic Force and Clearances

2011 ◽  
Vol 18 (1-2) ◽  
pp. 3-12 ◽  
Author(s):  
Jing Jiang ◽  
Qing-Kai Han ◽  
Chao-Feng Li ◽  
Hong-Liang Yao ◽  
Shu-Ying Liu
Keyword(s):  
Nonlinear Dynamic Analysis ◽  
Deep Understanding ◽  
Degree Of Freedom ◽  
The Novel ◽  
Hysteretic Model ◽  
Dynamical Equations ◽  
Motion Patterns ◽  
Dynamical Behaviors ◽  
Cone Crusher ◽  
Two Degree Of Freedom

Based on the analysis on crushing process and hysteresis of material layers, a hysteretic model with symmetrical clearances is presented. The mechanical model of two-degree of freedom with bilinear hysteresis and its dynamical equations of system are proposed. In order to further investigate the dynamic characteristics of the novel vibratory cone crusher, the system is also simplified into a dynamical system of single degree of freedom with a bilinear hysteretic component together with clearances. According to some nonlinear dynamic analysis tools such as bifurcation diagram, Lyapunov exponents, Poincare section, etc., different motion patterns of the system are discussed, including periodic, periodic doubling, chaos and other characteristics. These theoretical results will provide readers with deep understanding on the regular and complex dynamical behaviors of the vibratory cone crusher due to the hysteresis with clearances.

N-soliton solutions and localized wave interaction solutions of a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyamaf equation

2021 ◽  
pp. 2150277
Author(s):  
Hongcai Ma ◽  
Qiaoxin Cheng ◽  
Aiping Deng
Keyword(s):  
Dynamic Behavior ◽  
Soliton Solution ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Bilinear Transformation ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Ytsf Equation ◽  
Line Soliton

[Formula: see text]-soliton solutions are derived for a (3 + 1)-dimensional potential-Yu–Toda–Sasa–Fukuyama (YTSF) equation by using bilinear transformation. Some local waves such as period soliton, line soliton, lump soliton and their interaction are constructed by selecting specific parameters on the multi-soliton solutions. By selecting special constraints on the two soliton solutions, period and lump soliton solution can be obtained; three solitons can reduce to the interaction solution between period soliton and line soliton or lump soliton and line soliton under special parameters; the interaction solution among period soliton and two line solitons, or the interaction solution for two period solitons or two lump solitons via taking specific constraints from four soliton solutions. Finally, some images of the results are drawn, and their dynamic behavior is analyzed.

Generalized Darboux transformation and asymptotic analysis on the degenerate dark-bright solitons for a coupled nonlinear Schrödinger system

2021 ◽  
Author(s):  
He-yuan Tian ◽  
Bo Tian ◽  
Yan Sun ◽  
Su-Su Chen
Keyword(s):  
Asymptotic Analysis ◽  
Darboux Transformation ◽  
Coupled System ◽  
Bound State ◽  
Bright Soliton ◽  
Spectral Parameters ◽  
Nonlinear Superposition ◽  
Bright Solitons ◽  
Generalized Darboux Transformation ◽  
The Given

Abstract In this paper, our work is based on a coupled nonlinear Schr ̈odinger system in a two-mode nonlinear fiber. A (N,m)-generalized Darboux transformation is constructed to derive the Nth-order solutions, where the positive integers N and m denote the numbers of iterative times and of distinct spectral parameters, respectively. Based on the Nth-order solutions and the given steps to perform the asymptotic analysis, it is found that a degenerate dark-bright soliton is the nonlinear superposition of several asymptotic dark-bright solitons possessing the same profile. For those asymptotic dark-bright solitons, their velocities are z-dependent except that one of those velocities could become z-independent under the certain condition, where z denotes the evolution dimension. Those asymptotic dark-bright solitons are reflected during the interaction. When a degenerate dark-bright soliton interacts with a nondegenerate/degenerate dark-bright soliton, the interaction is elastic, and the asymptotic bound-state dark-bright soliton with z-dependent or z-independent velocity could take place under certain conditions. Our study extends the investigation on the degenerate solitons from the bright soliton case for the scalar equations to the dark-bright soliton case for a coupled system.

Inelastic Interaction of Double-Valley Dark Solitons for the Hirota Equation

2021 ◽  
Vol 38 (9) ◽  
pp. 090201
Author(s):  
Xiao-Man Zhang ◽  
Yan-Hong Qin ◽  
Li-Ming Ling ◽  
Li-Chen Zhao
Keyword(s):  
Inelastic Interaction ◽  
Hirota Equation ◽  
Dark Solitons

Mixed interaction solutions for the coupled nonlinear Schrödinger equations

2021 ◽  
pp. 2150004
Author(s):  
Yaning Tang ◽  
Jiale Zhou
Keyword(s):  
Darboux Transformation ◽  
Rogue Waves ◽  
Transformation Method ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Coupled Nonlinear Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Interaction Solutions ◽  
Nonlinear Schrodinger Equations

We investigate the mixed interaction solutions of the coupled nonlinear Schrödinger equations (CNLSE) through the Darboux transformation method. First of all, we derive the nonsingular localized wave solutions for two cases of CNLSE by the Darboux transformation method and matrix analysis method. Furthermore, we take a limit technique to deduce rogue waves and divide the rogue waves into four categories through analyzing their dynamic behaviors. Based on the obtained theorems, the Darboux transformations are presented to solve interaction solutions between distinct nonlinear waves. In this paper, we mainly study four types. Finally, the dynamic characteristics of the constructed these solutions are analyzed by sequences of interesting figures plotted with the help of Maple.

Lump-type solutions, interaction solutions and periodic wave solutions of a (3+1)-dimensional Korteweg–de Vries equation

2019 ◽  
Vol 33 (27) ◽  
pp. 1950319 ◽  
Author(s):  
Hongfei Tian ◽  
Jinting Ha ◽  
Huiqun Zhang
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Interaction Solutions ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Hirota Bilinear Form ◽  
De Vries ◽  
Hirota Bilinear

Based on the Hirota bilinear form, lump-type solutions, interaction solutions and periodic wave solutions of a (3[Formula: see text]+[Formula: see text]1)-dimensional Korteweg–de Vries (KdV) equation are obtained. The interaction between a lump-type soliton and a stripe soliton including two phenomena: fission and fusion, are illustrated. The dynamical behaviors are shown more intuitively by graphics.

scholarly journals Investigation of Interaction Solutions for Modified Korteweg-de Vries Equation by Consistent Riccati Expansion Method

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jin-Fu Liang ◽  
Xun Wang
Keyword(s):  
Vries Equation ◽  
Soliton Solutions ◽  
Wave Interaction ◽  
Expansion Method ◽  
Periodic Wave ◽  
Mkdv Equation ◽  
Jacobi Elliptic Function ◽  
Interaction Solutions ◽  
De Vries ◽  
Consistent Riccati Expansion

A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).

scholarly journals Lump-Type Wave and Interaction Solutions of the Bogoyavlenskii–Kadomtsev–Petviashvili Equation

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chuanjian Wang ◽  
Hui Fang
Keyword(s):  
Wave Solution ◽  
Solitary Wave Solutions ◽  
Kink Wave Solution ◽  
Type Wave ◽  
Interaction Solutions ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Petviashvili Equation ◽  
Kink Wave ◽  
Hermitian Quadratic Form

Lump-type wave solution of the Bogoyavlenskii–Kadomtsev–Petviashvili equation is constructed by using the bilinear structure and Hermitian quadratic form. The dynamical behaviors of lump-type wave solution are investigated and presented analytically and graphically. Furthermore, we discuss the interaction between a lump-type wave and a kink wave solution. Absorb-emit interaction between two kinds of solitary wave solutions is shown. This kind of interaction solution can be regarded as a lump-type wave which propagates on the kink wave background.

scholarly journals Evolution of a Random Directional Wave and Freak Wave Occurrence

2009 ◽  
Vol 39 (3) ◽  
pp. 621-639 ◽  
Author(s):  
Takuji Waseda ◽  
Takeshi Kinoshita ◽  
Hitoshi Tamura
Keyword(s):  
Wave Height ◽  
Wave Interaction ◽  
Northwest Pacific ◽  
Frequency Bandwidth ◽  
Extreme Wave ◽  
Spectral Parameters ◽  
Freak Wave ◽  
Resonant Wave ◽  
Directional Wave ◽  
Wind Sea

Abstract The evolution of a random directional wave in deep water was studied in a laboratory wave tank (50 m long, 10 m wide, 5 m deep) utilizing a directional wave generator. A number of experiments were conducted, changing the various spectral parameters (wave steepness 0.05 < ɛ < 0.11, with directional spreading up to 36° and frequency bandwidth 0.2 < δk/k < 0.6). The wave evolution was studied by an array of wave wires distributed down the tank. As the spectral parameters were altered, the wave height statistics change. Without any wave directionality, the occurrence of waves exceeding twice the significant wave height (the freak wave) increases as the frequency bandwidth narrows and steepness increases, due to quasi-resonant wave–wave interaction. However, the probability of an extreme wave rapidly reduces as the directional bandwidth broadens. The effective Benjamin–Feir index (BFIeff) is introduced, extending the BFI (the relative magnitude of nonlinearity and dispersion) to incorporate the effect of directionality, and successfully parameterizes the observed occurrence of freak waves in the tank. Analysis of the high-resolution hindcast wave field of the northwest Pacific reveals that such a directionally confined wind sea with high extreme wave probability is rare and corresponds mostly to a swell–wind sea mixed condition. Therefore, extreme wave occurrence in the sea as a result of quasi-resonant wave–wave interaction is a rare event that occurs only when the wind sea directionality is extremely narrow.

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