Construction of rogue waves and conservation laws of the complex coupled Kadomtsev–Petviashvili equation

2020 ◽  
Vol 34 (12) ◽  
pp. 2050115
Author(s):  
H. I. Abdel-Gawad ◽  
M. Tantawy ◽  
Mustafa Inc ◽  
A. Yusuf
Keyword(s):  
Conservation Laws ◽  
Extension Theorem ◽  
Rogue Waves ◽  
Contour Plot ◽  
Science And Engineering ◽  
Quadratic Invariant ◽  
Governing System ◽  
Petviashvili Equation ◽  
The Unified Method ◽  
Unified Method

In this work, the solutions of the real coupled Kadomtsev–Petviashvili equations (KPEs) are obtained and they are used to find the ones of the complex system. To this end, the extension theorem is used. The formation of rogue wave (RW) is shown to occur via inelastic collisions of lines-lump solitons and periodic waves and also of two elliptic waves. The quadratic invariant is used to show the phase portrait and bifurcation through the contour plot. Further, the conservation laws (CLs) are constructed by means of new conservation theorem by finding the governing system of equation. The results are given, work consolidated from a previous work, by the first two authors, for suggesting a mechanism for the formation of RW when the complex Korteweg–de Vries equation was studied. Based on the exact solutions found here, numerical computations are carried out and they are represented graphically. Finally, the unified method (UM) can be applied to solve a variety of partial differential equations in science and engineering.

scholarly journals The unified method for abundant soliton solutions of local time fractional nonlinear evolution equations

2021 ◽  
Vol 22 ◽  
pp. 103979
Author(s):  
Nauman Raza ◽  
Muhammad Hamza Rafiq ◽  
Melike Kaplan ◽  
Sunil Kumar ◽  
Yu-Ming Chu
Keyword(s):  
Local Time ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
The Unified Method ◽  
Unified Method

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.

scholarly journals Lie analysis, conservation laws and travelling wave structures of nonlinear Bogoyavlenskii–Kadomtsev–Petviashvili equation

2020 ◽  
Vol 19 ◽  
pp. 103492
Author(s):  
Adil Jhangeer ◽  
Amjad Hussain ◽  
M. Junaid-U-Rehman ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Conservation Laws ◽  
Travelling Wave ◽  
Petviashvili Equation

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.

Rogue waves in multiple-solitons–inelastic collisions — The complex Sharma–Tasso–Olver equation

2018 ◽  
Vol 32 (08) ◽  
pp. 1750360 ◽  
Author(s):  
H. I. Abdel-Gawad ◽  
M. Tantawy
Keyword(s):  
Spectral Band ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Inelastic Collisions ◽  
Periodic Waves ◽  
Mechanism Of Formation ◽  
Free Parameters ◽  
Generalized Unified Method ◽  
The Generalized Unified Method ◽  
Unified Method

Very recently, a mechanism to the formation of rogue waves (RWs) has been proposed by the authors. In this paper, the formation of RWs in case of the complex Sharma–Tasso–Olver (STO) equation is studied. In the STO equation, one, two and three-soliton solutions are obtained. Due to the inelastic collisions, these soliton waves are fused to one. Under the free parameters constraint this behavior do occurs. The mechanism of formation of RWs is due to the collisions of solitons and multi-periodic waves (like spectral band). These RWs as giant waves, which may be very sharp or chaotic are similar to RWs in laser. The work is done here by using the generalized unified method (GUM).

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