Rational solutions and rogue waves of the generalized (2+1)-dimensional Kadomtsev–Petviashvili equation

2022 ◽  
Author(s):  
Yingying Xie ◽  
Yongsheng Yan ◽  
Lingfei Li
Keyword(s):  
Rogue Waves ◽  
Rational Solutions ◽  
Petviashvili Equation

Dynamics of rational solutions in a new generalized Kadomtsev–Petviashvili equation

2019 ◽  
Vol 33 (35) ◽  
pp. 1950437 ◽  
Author(s):  
K. Hosseini ◽  
M. Aligoli ◽  
M. Mirzazadeh ◽  
M. Eslami ◽  
J. F. Gómez-Aguilar
Keyword(s):  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Rogue Waves ◽  
Kp Equation ◽  
Rational Solutions ◽  
Research Papers ◽  
Petviashvili Equation ◽  
Special Cases ◽  
The Subject ◽  
Generalized Kp Equation

Rational solutions of nonlinear evolution (NLE) equations have been the subject of numerous research papers. In this paper, a new generalized Kadomtsev–Petviashvili (KP) equation with diverse applications is investigated analytically. Multiple solitons, breather and rogue waves, and complexitons as special cases of rational solutions to the new generalized KP equation are formally extracted with the help of symbolic computations. Some two- and three-dimensional figures are considered to show the dynamics of rational solutions in the new generalized KP equation.

Families of exact solutions of the generalized (3+1)-dimensional nonlinear-wave equation

2018 ◽  
Vol 32 (29) ◽  
pp. 1850359 ◽  
Author(s):  
Wenhao Liu ◽  
Yufeng Zhang
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Lump Solution ◽  
Rational Solutions ◽  
Bilinear Method ◽  
The One ◽  
Wave Limit

In this paper, the traveling wave method is employed to investigate the one-soliton solutions to two different types of bright solutions for the generalized (3[Formula: see text]+[Formula: see text]1)-dimensional nonlinear-wave equation, primarily. In the following parts, we derive the breathers and rational solutions by using the Hirota bilinear method and long-wave limit. More specifically, we discuss the lump solution and rogue wave solution, in which their trajectory will be changed by varying the corresponding coefficient or coordinate axis. On the one hand, the breathers express the form of periodic line waves in different planes, on the other hand, rogue waves are localized in time.

Analytic rogue wave-type solutions for the generalized (2+1)-dimensional Boussinesq Equation

2021 ◽  
pp. 2150380
Author(s):  
Xiu-Rong Guo
Keyword(s):  
Boussinesq Equation ◽  
Rogue Wave ◽  
Wave Interaction ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Wave Type ◽  
Hirota Bilinear Form ◽  
Basic Facts ◽  
2D And 3D ◽  
Solitary Wave Interaction

Based on the Hirota bilinear form of the generalized (2+1)-dimensional Boussinesq equation, which can be expressed as the shallow water wave mechanism appearing in fluid mechanics, we applied the new polynomial functions to construct the rational solutions and rogue wave-type solutions. Next, the system parameters control on the rational solutions and rogue wave-type solutions were also shown. As a result, we found the following basic facts: (i) these parameters may affect the wave shapes, amplitude, and bright/dark for this considered equation, (ii) the solitary wave interaction rogue waves and triplet rogue wave-type solutions can be viewed on [Formula: see text], [Formula: see text], and [Formula: see text] planes, respectively. Their nonlinear dynamic behaviors were presented by numerical simulation of the 2D- and 3D-plots.

Characteristics of the breather waves, rogue waves and solitary waves in an extended (3 + 1)-dimensional Kadomtsev–Petviashvili equation

2019 ◽  
Vol 29 (8) ◽  
pp. 2964-2976
Author(s):  
Hui Wang ◽  
Shou-Fu Tian ◽  
Yi Chen
Keyword(s):  
Solitary Waves ◽  
Dynamical Behavior ◽  
Rogue Waves ◽  
Graphical Analysis ◽  
Nonlinear Phenomena ◽  
Kp Equation ◽  
Bilinear Method ◽  
Test Technique ◽  
Content Type ◽  
Petviashvili Equation

Purpose The purpose of this paper is to study the breather waves, rogue waves and solitary waves of an extended (3 + 1)-dimensional Kadomtsev–Petviashvili (KP) equation, which can be used to depict many nonlinear phenomena in fluid dynamics and plasma physics. Design/methodology/approach The authors apply the Bell’s polynomial approach, the homoclinic test technique and Hirota’s bilinear method to find the breather waves, rogue waves and solitary waves of the extended (3 + 1)-dimensional KP equation. Findings The results imply that the extended (3 + 1)-dimensional KP equation has breather wave, rogue wave and solitary wave solutions. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Originality/value These results may help us to further study the local structure and the interaction of solutions in KP-type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of such equations.

Localized waves and interaction solutions to an extended (3+1)-dimensional Kadomtsev–Petviashvili equation

2020 ◽  
Vol 34 (06) ◽  
pp. 2050076 ◽  
Author(s):  
Han-Dong Guo ◽  
Tie-Cheng Xia ◽  
Wen-Xiu Ma
Keyword(s):  
Three Dimensional ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Dark Soliton ◽  
Complex Conjugate ◽  
Bilinear Method ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Parameter Constraints ◽  
Localized Waves

In this paper, an extended (3[Formula: see text]+[Formula: see text]1)-dimensional Kadomtsev–Petviashvili (KP) equation is studied via the Hirota bilinear derivative method. Soliton, breather, lump and rogue waves, which are four types of localized waves, are obtained. N-soliton solution is derived by employing bilinear method. Then, line or general breathers, two-order line or general breathers, interaction solutions between soliton and line or general breathers are constructed by complex conjugate approach. These breathers own different dynamic behaviors in different planes. Taking the long wave limit method on the multi-soliton solutions under special parameter constraints, lumps, two- and three-lump and interaction solutions between dark soliton and dark lump are constructed, respectively. Finally, dark rogue waves, dark two-order rogue waves and related interaction solutions between dark soliton and dark rogue waves or dark lump are also demonstrated. Moreover, dynamical characteristics of these localized waves and interaction solutions are further vividly demonstrated through lots of three-dimensional graphs.

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.

Breather wave, rogue wave and lump wave solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

2018 ◽  
Vol 32 (20) ◽  
pp. 1850223 ◽  
Author(s):  
Ming-Zhen Li ◽  
Bo Tian ◽  
Yan Sun ◽  
Xiao-Yu Wu ◽  
Chen-Rong Zhang
Keyword(s):  
Wave Velocity ◽  
Water Waves ◽  
Rogue Wave ◽  
Rogue Waves ◽  
Dispersion Effect ◽  
Wave Solutions ◽  
Weak Perturbation ◽  
Petviashvili Equation ◽  
Nonlinearity Effect ◽  
Lump Wave

Under investigation in this paper is a (3[Formula: see text]+[Formula: see text]1)-dimensional generalized Kadomtsev–Petviashvili equation, which describes the long water waves and small-amplitude surface waves with the weak nonlinearity, weak dispersion and weak perturbation in a fluid. Via the Hirota method and symbolic computation, the lump wave, breather wave and rogue wave solutions are obtained. We graphically present the lump waves under the influence of the dispersion effect, nonlinearity effect, disturbed wave velocity effects and perturbed effects: Decreasing value of the dispersion effect can lead to the range of the lump wave decreases, but has no effect on the amplitude. When the value of the nonlinearity effect or disturbed wave velocity effects increases respectively, lump wave’s amplitude decreases but lump wave’s location keeps unchanged. Amplitudes of the lump waves are independent of the perturbed effects. Breather waves and rogue waves are displayed: Rogue waves emerge when the periods of the breather waves go to the infinity. When the value of the dispersion effect decreases, range of the rogue wave increases. When the value of the nonlinearity effect or disturbed wave velocity effects decreases respectively, rogue wave’s amplitude decreases. Value changes of the perturbed effects cannot influence the rogue wave.

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.

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