scholarly journals Line Soliton Interactions for Shallow Ocean Waves and Novel Solutions with Peakon, Ring, Conical, Columnar, and Lump Structures Based on Fractional KP Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bo Xu ◽  
Yufeng Zhang ◽  
Sheng Zhang
Keyword(s):  
Rational Solution ◽  
Soliton Solutions ◽  
Ocean Waves ◽  
Columnar Structure ◽  
Lump Solution ◽  
Spatial Structures ◽  
Kp Equation ◽  
Spreading Process ◽  
Soliton Interactions ◽  
Special Case

It is well known that the celebrated Kadomtsev-Petviashvili (KP) equation has many important applications. The aim of this article is to use fractional KP equation to not only simulate shallow ocean waves but also construct novel spatial structures. Firstly, the definitions of the conformable fractional partial derivatives and integrals together with a physical interpretation are introduced and then a fractional integrable KP equation consisting of fractional KPI and KPII equations is derived. Secondly, a formula for the fractional n -soliton solutions of the derived fractional KP equation is obtained and fractional line one-solitons with bend, wavelet peaks, and peakon are constructed. Thirdly, fractional X-, Y- and 3-in-2-out-type interactions in the fractional line two- and three-soliton solutions of the fractional KPII equation are simulated for shallow ocean waves. Besides, a falling and spreading process of a columnar structure in the fractional line two-soliton solution is also simulated. Finally, a fractional rational solution of the fractional KP equation is obtained including the lump solution as a special case. With the development of time, the nonlinear dynamic evolution of the fractional lump solution of the fractional KPI equation can change from ring and conical structures to lump structure.

Two–Soliton Solutions Of The Kadomtsevpetviashvili Equation

2012 ◽  
Author(s):  
Wei King Tiong ◽  
Chee Tiong Ong ◽  
Mukheta Isa
Keyword(s):  
Analytic Solution ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Kp Equation ◽  
Bilinear Method ◽  
Traditional Group ◽  
Soliton Interactions ◽  
De Vries ◽  
Hirota Bilinear

Beberapa keputusan tentang penjanaan penyelesaian soliton oleh persamaan Kadomtsev–Petviashvili akan dibincangkan dalam kertas ini. Kaedah teori kumpulan mampu memberikan penyelesaian secara analitik kerana persamaan KP mempunyai ketakterhinggaan banyaknya hukum keabadian. Dengan kaedah Bilinear Hirota, ditunjukkan melalui simulasi berkomputer bagaimana penyelesaian dua soliton persamaan KP mampu menghasilkan strukturstruktur “triad”, kuadruplet dan struktur tak beresonan dalam interaksi soliton. Kata kunci: Soliton, kaedah Bilinear Hirota, persamaan Kortewegde Vries dan Kadomtsev- Petviashvili Several findings on soliton solutions generated by the Kadomtsev–Petviashvili (KP) equation were discussed in this paper. This equation is a two dimensional of the Korteweg–de Vries (KdV) equation. Traditional group–theoretical approach can generate analytic solution of solitons because KP equation has infinitely many conservation laws. By using Hirota Bilinear method, we show via computer simulation how two solitons solution of KP equation produces triad, quadruplet and a non–resonance structures in soliton interactions. Key words: Soliton, Hirota Bilinear method, Korteweg-de Vries and Kadomtsev-Petviashvili equations

scholarly journals Tropical limit of matrix solitons and entwining Yang–Baxter maps

2020 ◽  
Vol 110 (11) ◽  
pp. 3015-3051
Author(s):  
Aristophanes Dimakis ◽  
Folkert Müller-Hoissen
Keyword(s):  
Soliton Solution ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Baxter Equation ◽  
Kp Equation ◽  
Lattice Equation ◽  
Soliton Interactions ◽  
The Matrix ◽  
Tropical Limit ◽  
Toda Lattice Equation

Abstract We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

scholarly journals Group analysis and conservation laws of an integrable Kadomtsev–Petviashvili equation

2018 ◽  
Vol 24 (1) ◽  
pp. 34-46
Author(s):  
Gangwei Wang ◽  
Qi Wang ◽  
Yingwei Chen
Keyword(s):  
Conservation Laws ◽  
Vector Fields ◽  
Group Analysis ◽  
Soliton Solutions ◽  
Symmetry And Conservation Laws ◽  
Kp Equation ◽  
Infinitesimal Generators ◽  
Petviashvili Equation ◽  
Power Series Solutions ◽  
Special Case

In this paper, an integrable KP equation is studied using symmetry and conservation laws. First, on the basis of various cases of coefficients, we construct the infinitesimal generators. For the special case, we get the corresponding geometry vector fields, and then from known soliton solutions we derive new soliton solutions. In addition, the explicit power series solutions are derived. Lastly, nonlinear self-adjointness and conservation laws are constructed with symmetries.

Two integrable extensions of the Kadomtsev-Petviashvili equation

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Kp Equation ◽  
Completely Integrable ◽  
Multiple Soliton Solutions ◽  
Petviashvili Equation ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

AbstractIn this work, two new completely integrable extensions of the Kadomtsev-Petviashvili (eKP) equation are developed. Multiple soliton solutions and multiple singular soliton solutions are derived to demonstrate the compatibility of the extensions of the KP equation.

Abundant Lump Solution and Interaction Phenomenon of (3+1)-Dimensional Generalized Kadomtsev–Petviashvili Equation

2019 ◽  
Vol 20 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jianqing Lü ◽  
Sudao Bilige ◽  
Xiaoqing Gao
Keyword(s):  
Symbolic Computation ◽  
Bilinear Form ◽  
Lump Solution ◽  
Lump Solutions ◽  
Interaction Solution ◽  
Interaction Solutions ◽  
Petviashvili Equation ◽  
Contour Plots ◽  
Special Case ◽  
Interaction Phenomenon

AbstractIn this paper, with the help of symbolic computation system Mathematica, six kinds of lump solutions and two classes of interaction solutions are discussed to the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation via using generalized bilinear form with a dependent variable transformation. Particularly, one special case are plotted as illustrative examples, and some contour plots with different determinant values are presented. Simultaneously, we studied the trajectory of the interaction solution.

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.

SOLITON INTERACTIONS FOR THE GENERALIZED (3+1)-DIMENSIONAL BOUSSINESQ EQUATION

2012 ◽  
Vol 26 (07) ◽  
pp. 1250062 ◽  
Author(s):  
XIAO-LING GAI ◽  
YI-TIAN GAO ◽  
XIN YU ◽  
ZHI-YUAN SUN
Keyword(s):  
Soliton Solution ◽  
Boussinesq Equation ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Propagation Process ◽  
Soliton Propagation ◽  
Soliton Interactions ◽  
Pass Through ◽  
The Moment ◽  
The One

Generalized (3+1)-dimensional Boussinesq equation is investigated in this paper. Through the dependent variable transformation and symbolic computation, the one- and two-soliton solutions are obtained. With the one-soliton solution, the coefficient effects in the soliton propagation process are investigated. Through analyzing the two-soliton solution, two kinds of two-soliton interactions are presented: (i) Two solitons merge into a bigger one whose amplitude increases but does not exceed the sum of the two at the moment of the collision; (ii) Two solitons can pass through each other, and their shapes keep unchanged with a phase shift after the separation. In addition, two kinds of analytic solutions are discussed: (i) "Amplitudes" of the two analytic solutions immediately turn to negative (positive) infinity after the "collision"; (ii) Two analytic solutions are fused into a higher peak (valley) at the moment of "collision", whose "amplitudes" change to negative (positive) infinity after the separation.

Extended KP equations and extended system of KP equations: multiple-soliton solutions

2011 ◽  
Vol 89 (7) ◽  
pp. 739-743 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Dispersion Relations ◽  
Kp Equation ◽  
Extended System ◽  
Multiple Soliton Solutions ◽  
Simplified Method ◽  
Kp Equations

In this work we study an extended Kadomtsev–Petviashvili (KP) equation and a system of KP equations. We show that the extension terms do not kill the integrability of typical models. Hereman’s simplified method is used to justify this goal. Multiple soliton solutions will be derived for each model. The analysis highlights the effects of the extension terms on the dispersion relations, and hence on the structures of the solutions.

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