scholarly journals Bilinear Form, Soliton, Breather, Hybrid and Periodic-Wave Solutions for a (3+1)-Dimensional Korteweg-De Vries Equation in a Fluid

2021 ◽  
Author(s):  
Cheng Chong-Dong ◽  
Tian Bo ◽  
Zhang Chen-Rong ◽  
Zhao Xin
Keyword(s):  
Bilinear Form ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Second Order ◽  
Periodic Wave Solutions ◽  
First Order ◽  
Wave Solutions ◽  
De Vries ◽  
Limiting Condition

Abstract Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg-de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe that the interaction between two perpendicular first-order breathers on the x-y and x-z planes and the periodic line wave interacts with the first-order breather on the y-z plane, where x, y and z are the independent variables in the equation. Furthermore, we discuss the effects of α, β, γ and δ on the amplitudes of the second-order breathers, where α, β, γ and δ are the constant coefficients in the equation: Amplitude of the second-order breather decreases as α increases; Amplitude of the second-order breather increases as β increases; Amplitude of the second-order breather keeps invariant as γ and δ increase. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions. Furthermore, we find that the periodic-wave solutions approach to the one-soliton solutions under certain limiting condition.

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.

FRACTAL AND CHAOTIC PATTERNS OF GENERAL KORTEWEG–DE VRIES SYSTEM IN (2+1)-DIMENSIONS

2006 ◽  
Vol 20 (28) ◽  
pp. 4843-4854 ◽  
Author(s):  
CHUN-LONG ZHENG ◽  
HAI-PING ZHU ◽  
JIAN-PING FANG
Keyword(s):  
Soliton Solutions ◽  
Periodic Wave ◽  
Analytical Investigation ◽  
Variable Separation ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Solitary Solutions ◽  
De Vries ◽  
Localized Patterns ◽  
Variable Separation Approach

With the aid of an extended projective method and a variable separation approach, new families of variable separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) with arbitrary functions for (2+1)-dimensional general Korteweg–de Vries (GKdV) system are derived. Analytical investigation of the (2+1)-dimensional GKdV system shows the existence of abundant stable localized coherent excitations such as dromions, lumps, peakons, compactons and ring soliton solutions as well as rich fractal and chaotic localized patterns in terms of the derived solitary solutions or the variable separation solutions when we consider appropriate boundary conditions and/or initial qualifications.

scholarly journals Mapping Methods to Solve a Modified Korteweg-de Vries Type Equation

2015 ◽  
Vol 20 (2) ◽  
pp. 42 ◽  
Author(s):  
Edamana. V. Krishnan
Keyword(s):  
Vries Equation ◽  
Type Equation ◽  
Elliptic Functions ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Hyperbolic Functions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Singular Wave

In this paper, we employ mapping methods to construct exact travelling wave solutions for a modified Korteweg-de Vries equation. We have derived periodic wave solutions in terms of Jacobi elliptic functions, kink solutions and singular wave solutions in terms of hyperbolic functions.  

Quasi-periodic wave solutions and asymptotic properties for a fifth-order Korteweg–de Vries type equation

2016 ◽  
Vol 30 (18) ◽  
pp. 1650223 ◽  
Author(s):  
Chun-Yan Qin ◽  
Shou-Fu Tian ◽  
Xiu-Bin Wang ◽  
Tian-Tian Zhang
Keyword(s):  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Ocean Dynamics ◽  
Nonlinear Phenomena ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Fifth Order

Under investigation in this paper is a fifth-order Korteweg–de Vries type (fKdV-type) equation with time-dependent coefficients, which can be used to describe many nonlinear phenomena in fluid mechanics, ocean dynamics and plasma physics. The binary Bell polynomials are employed to find its Hirota’s bilinear formalism with an extra auxiliary variable, based on which its [Formula: see text]-soliton solutions can be also directly derived. Furthermore, by considering multi-dimensional Riemann theta function, a lucid and straightforward generalization of the Hirota–Riemann method is presented to explicitly construct the multiperiodic wave solutions of the equation. Finally, the asymptotic properties of these periodic wave solutions are strictly analyzed to reveal the relationships between periodic wave solutions and soliton solutions.

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.

Resonance Y-type soliton, hybrid and quasi-periodic wave solutions of a generalized (2+1)-dimensional nonlinear wave equation

2021 ◽  
Author(s):  
Lingchao He ◽  
Jianwen Zhang ◽  
Zhonglong Zhao
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Water Waves ◽  
Asymptotic Properties ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Before And After

Abstract In this paper, we consider a generalized (2+1)-dimensional nonlinear wave equation. Based on the bilinear, the N-soliton solutions are obtained. The resonance Y-type soliton and the interaction solutions between M-resonance Y-type solitons and P-resonance Y-type solitons are constructed by adding some new constraints to the parameters of the N-soliton solutions. The new type of two-opening resonance Y-type soliton solutions are presented by choosing some appropriate parameters in 3-soliton solutions. The hybrid solutions consisting of resonance Y-type solitons, breathers and lumps are investigated. The trajectories of the lump waves before and after the collision with the Y-type solitons are analyzed from the perspective of mathematical mechanism. Furthermore, the multi-dimensional Riemann-theta function is employed to investigate the quasi-periodic wave solutions. The one-periodic and two-periodic wave solutions are obtained. The asymptotic properties are systematically analyzed, which establish the relations between the quasi-periodic wave solutions and the soliton solutions. The results may be helpful to provide some effective information to analyze the dynamical behaviors of solitons, fluid mechanics, shallow water waves and optical solitons.

Nonlocal Symmetry and Consistent Tanh Expansion Method for the Coupled Integrable Dispersionless Equation

2016 ◽  
Vol 71 (3) ◽  
pp. 235-240 ◽  
Author(s):  
Hengchun Hu ◽  
Xiao Hu ◽  
Bao-Feng Feng
Keyword(s):  
Soliton Solutions ◽  
Expansion Method ◽  
Periodic Wave ◽  
Cnoidal Wave ◽  
Nonlocal Symmetry ◽  
Nonlocal Symmetries ◽  
Periodic Wave Solutions ◽  
Wave Solutions

AbstractNonlocal symmetries are obtained for the coupled integrable dispersionless (CID) equation. The CID equation is proved to be consistent, tanh-expansion solvable. New, exact interaction excitations such as soliton–cnoidal wave solutions, soliton–periodic wave solutions, and multiple resonant soliton solutions are discussed analytically and shown graphically.

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).

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